STUDIES IN
DEDUCTIVE LOGIC
a
Tl c
B
HAG GNANTIAI OVDIC
TIC YneNANTIAI OVH
STUDIES
IN
DEDUCTIVE LOGIC
SL ^anual for >tutient0
BY
W. STANLEY JEVONS
LL.D. (EDINB.)) M.A. (LOND.), F.R.S.
SECOND EDITION
3Lonton
MACMILLAN AND CO.
1884
The right of translation and reproduction is reserved.
ELECTRONIC VERSION
AVAILABLE
Printed by R. & R. CLARK, Edinburgh.
PREFACE.
IN preparing these Studies I have tried to carry forward
the chief purpose of my Elementary Lessons in Logic, which
purpose was the promotion of practical training in Logic.
In the preface to those Lessons I said in 1870: The
relations of propositions and the forms of argument present
as precise a subject of instruction and as vigorous an exer
cise of thought, as the properties of geometrical figures or
the rules of Algebra. Yet every schoolboy is made to learn
mathematical problems which he will never employ in after
life, and is left in total ignorance of those simple principles
and forms of reasoning which will enter into the thoughts of
every hour. ... In my own classes I have constantly
found that the working and solution of logical questions,
the examination of arguments and the detection of fallacies,
is a not less practicable and useful exercise of mind than
is the performance of calculations and the solution of
problems in a mathematical class.
The considerable use which has been made of the
Elementary Lessons seems to show that they meet an educa
tional want of the present day. The time has now perhaps
viii PREFACE.
arrived when facilities for a more thorough course of logical
training may be offered to teachers and students.
For a long time back there have been published books
containing abundance of mathematical exercises, and not
a few works consist exclusively of such exercises. In
recent years the teachers of other branches of science,
such as Chemistry and the Theory of Heat, have been
furnished with similar collections of problems and numerical
examples. There can be no doubt about the value of such
exercises when they can be had. The great point in
education is to throw the mind of the learner into an active,
instead of a passive state. It is of no use to listen to a
lecture or to read a lesson unless the mind appropriates
and digests the ideas and principles put before it. The
working of problems and the answering of definite questions
is the best, if not almost the only, means of ensuring this
active exercise of thought. It is possible that at Cambridge
mathematical gymnastics have been pushed to an extreme,
the study of the principles and philosophy of Mathematics
being almost forgotten in the race to solve the greatest
possible number of the most difficult problems in the
shortest possible time. But there can be no manner of
doubt that from the simple addition sums of the schoolboy
up to problems in the Calculus of Variations and the
Theory of Probability, the real study of Mathematics must
consist in the student cracking his own nuts, and gaming
for himself the kernel of understanding.
So it must be in Logic. Students of Logic must have
logical nuts to crack. Opinions may differ, indeed, as to
PREFACE. ix
the value of logical training in any form. That value is
twofold, arising both from the general training of the mental
powers and from the command of reasoning processes
eventually acquired. I maintain that in both ways Logic,
when properly taught, need not fear comparison with the
Mathematics, and in the second point of view Logic is
decidedly superior to the sciences of quantity. Many
students acquire a wonderful facility in integrating differen
tial equations, and cracking other hard mathematical nuts,
who will never need to solve an equation again, after they
settle down in the conveyancer s chambers or the vicar s
parsonage. With the ordinary forms of logical inference
and of logical combination they will ceaselessly deal for
the rest of their lives ; yet for the knowledge of the forms
and principles of reasoning they generally trust to the light
of nature.
I do not deny that a mind of first-rate ability has con
siderable command of natural logic, which is often greatly
improved by a severe course of mathematical study. But
I have had abundant opportunities, both as a teacher and
an examiner, of estimating the logical facility of minds of
various training and capacity, and I have often been
astonished at the way in which even well-trained students
break down before a simple logical problem. A man who
is very ready at integration begins to hesitate and flounder
when he is asked such a simple question as the following :
If all triangles are plane figures, what information, if any,
does this proposition give us concerning things which are
not triangles? As to untrained thinkers, they seldom
x PREFACE.
discriminate between the most widely distinct assertions.
De Morgan has remarked in more than one place l that
a beginner, when asked what follows from Every A is B,
answers Every B is A of course The fact that such a
converse is often true in geometry, although it cannot be
inferred by pure logic, tends to mystify the student. Al
though all mathematical reasoning must necessarily be
logical if it be correct, yet the conditions of quantitative
reasoning are often such as actually to mislead the reasoner
who confuses them with the conditions of argumentation in
ordinary life. A mathematical education requires, in short,
to be corrected and completed, if indeed it should not be
preceded, by a logical education. There was never a
greater teacher of mathematics than De Morgan ; but from
his earliest essay on the Study of Mathematics to his very
latest writings, he always insisted upon the need of logical
as well as purely mathematical training. This was the
purpose of his tract of 1839, entitled, First Notions of Logic
preparatory to the Study of Geometry, subsequently reprinted
as the first chapter of the Formal Logic. A like idea
inspired his valuable essays On the Method of Teaching
Geometry, quoted above.
1 The Schoolmaster : Essays on Practical Education, 1836, vol. ii.
p. 1 20, note. This excellent essay On the Method of Teaching
Geometry was originally printed in the Quarterly Journal of Education,
No. XI. 1833, vol. vi. pp. 237-251. Similar views are put forth
in De Morgan s earlier work, On the Stiidy and Difficulties of Mathe
matics, published in 1831 by the Society for the Diffusion of Useful
Knowledge. See chapter xiv. See also De Morgan s Fourth Memoir
on the Syllogism, p. 4, in the Cambridge Philosophical Transactions
for 1860.
PREFACE. xi
Professor Sylvester, indeed, in his most curious tractate
upon the Laws of Verse (p. 19), has called in question the
nut -bearing powers of logic, saying : It seems to me
absurd to suppose that there exists in the science of pure
logic anything that bears a resemblance to the infinitely
developable and interminable euristic processes of mathe
matical science. To such a remark this volume is
perhaps the best possible answer, especially when it is
stated that I have had great difficulty in selecting and
compressing my materials so as to get them into a
volume of moderate size. If any person who thinks with
Professor Sylvester should object to the greater part of
the problems as dealing with concrete logic, let him look
to the end of this book, where he will find that the closely
printed Logical Index to the forms of law governing the
combinations of only three terms, fills four pages, without
in any way including the almost infinitely various logical
equivalents of those distinct forms. He will also learn that
a similarly complete index of the forms of logical law
governing the combinations of only five logical terms would
fill a library of 65,536 volumes. Surely there is scope
enough here for euristic processes.
An anxious and difficult task which I had to encounter in
compiling this book consisted in choosing the system or
systems of logical notation and method which were to be
expounded. When once the convenient but tyrannical
uniformity of the Aristotelian logic was overthrown, each
writer on the science proceeded to invent a new set of
xii PREFACE.
symbols. But it is impossible to employ alike the Greek
letters of Archbishop Thomson, the mysterious spiculae
of De Morgan, the cumbrous strokes, wedges, and dots of
Sir W. Hamilton, and the intricate mathematical formulae
of Boole. After a careful renewed study of the writings
of these eminent logicians I felt compelled in the first
place to discard the diverse and complicated notative
methods of De Morgan. Few or none admire more than
I do the extraordinary ingenuity, fertility, and, in a certain
way, the accuracy of De Morgan s logical writings. My
general indebtedness, both to those writings and to his
own unrivalled oral teaching, cannot be sufficiently ac
knowledged. I have, moreover, drawn many particular
hints from his works too numerous to be specified.
Nevertheless, to import his mysterious spiculae into
this book was to add a needless stumbling-block. The
question would have arisen too, which of his various
systems to adopt; for De Morgan created six equally
important concurrent syllogistic systems, the initial letters
of the names of which he characteristically threw into the
anagrams, Rue not! True? No! These systems were
the Relative, Undecided, Exemplar, Numerical, Onymatic,
and Transposed. See A Budget of Paradoxes, pp. 202-3.
There was in fact an unfortunate want of power of general
isation in De Morgan; his mind could dissect logical
questions into their very atoms, but he could not put the
particles of thought together again into a real system. As
his great antagonist, Sir W. Hamilton, remarked, De
Morgan was wanting in Architectonic Power.
PREFACE. xiii
It seems equally impossible, however, to adopt Sir
W. Hamilton s own logical symbols. His chief method
of notation has been briefly described in the Elementary
Lessons in Logic (p. 189). He also constructed or con
templated other systems of notation, as stated in his
Lectures on Logic (vol. iv., pp. 464476). In no case
do these notations seem to be so good as the earlier and
simpler one of Mr. George Bentham. And after a
laborious reinvestigation, rendered indispensable by the
composition of various parts of this book, I have been
forced to the conviction that in almost every case where
Hamilton differed from contemporaries or predecessors
he blundered. He was, as his admirers said, to put the
keystone into the arch of the Aristotelic syllogism ; but,
in spite of his Architectonic Power I fear we must
allow that his arch has collapsed. (See pp. 129133,
151-4, and 157-8, of this book.)
With the logical innovations of Dr. Thomson the case
is different. While he appears to enjoy the credit of an
independent discovery of the Quantification of the
Predicate, prior to any public and explicit statement of
the same by Hamilton, De Morgan, or Boole, but
posterior to the neglected work of Mr. George Bentham,
he did not commit the blunders of Hamilton, nor overlay
his work with useless crowds of short-hand symbols. He
most aptly completed the ancient scholastic notation of
propositions (A, E, I, O) by adding U, Y, ^ and w to
denote the new forms derived from Quantification of the
Predicate, carefully showing at the same time that ?? and
b
xiv PREFACE.
w are practical nonentities. I have therefore used his
notation for quantified propositions and syllogisms where
necessary.
Boole s great works are of course the foundation of almost
all subsequent progress in formal logic. My own views,
as I long since explicitly stated, 1 are moulded out of his.
Believing, however, that the mathematical dress into which
he threw his discoveries is not proper to them, and that
his quasi -mathematical processes are vastly more compli
cated than they need have been, I have of course preferred
my simpler version. Students who wish to comprehend
Boole s power and Boole s methods must go to the original
writings. It is really impossible that any abstract or
summary can give an adequate idea of the stupendous
efforts which Boole made to construct a general mathe
matical calculus of inference. Dr. Macfarlane, of Edinburgh,
has lately published a new version of Boole s system
under the title Algebra of Logic, but I am unable as
yet to discover that he has made any improvement on
Boole.
The writings of M. Delboeuf on Algorithmic Logic,
first printed in the Revue Philosophique for 1876, and since
reprinted, are very interesting, but were written in ignorance
of what had been done in this country by Boole and
others.
Quite recently Mr. Hugh MacColl, B.A., has published in
the Proceedings of the London Mathematical Society, and in
Mind, several papers upon a Calculus of Equivalent
1 Pure Logic, 1864, p. 3, etc.
PREFACE. xv
Statements, which arose out of an earlier article in the
Educational Times ^ His Calculus differs in several points
both from that of Boole and from that described in this
book as Equational Logic. Mr. MacColl rejects equations
in favour of implications ; thus my A = AB becomes with
him A : B, or A implies B. Even his letter-terms differ
in meaning from mine, since his letters denote propositions,
not things. Thus A : B asserts that the statement A
implies the statement B, or that whenever A is true, B is
also true. It is difficult to believe that there is any
advantage in these innovations; certainly, in preferring
implications to equations, Mr. MacColl ignores the necessity
of the equation for the application of the Principle of
Substitution. His proposals seem to me to tend towards
throwing Formal Logic back into its ante-Boolian con
fusion.
In one point, no doubt, his notation is very elegant,
namely, in using an accent as a sign of negation. A is
the negative of A j and as this accent can be applied with
the aid of brackets to terms of any degree of complexity,
there may sometimes be convenience in using it. Thus
(A + B) = A B ; (ABCD ...) = A + B + C +
D + . . . . I shall occasionally take the liberty of using
the accent in this way (see p. 199), but it is not often
needed. In the case of single negative terms, I find ex
perimentally that De Morgan s Italic negatives are the best.
The Italic a is not only far more clearly distinguished from
A than is A , but it is written with one pen-stroke less,
1 August 1871, also July 1877.
xvi PREFACE.
which in the long run is a matter of importance. The
student, of course, can use A for a whenever he finds it
convenient.
The logical investigations of Mr. A. J. Ellis, F.R.S., require
notice, because they are closely analogous to, if not nearly
identical with, my own. I am much indebted to him for
assisting me to become acquainted with his views. Not
only has he supplied me with an unpublished reprint, with
additions, of his articles in the Educational Times, but he
has allowed me access to the manuscripts of two elaborate
memoirs which he presented to the Royal Society, and
which are now preserved in the archives of the Society.
Some account of these investigations will be found in the
Proceedings of the Royal Society for April 1872, No. 134,
vol. xx. p. 307, and November 1873, vol. xxi. p. 497.
In the former place Mr. Ellis remarks : The above con
tributions are believed to be entirely original .... Jevons
first led my thoughts in this direction, but all resemblance
between us is entirely superficial. The question of resem
blance thus raised by Mr. Ellis must be left to others to
decide; but in order to avoid possible misapprehension,
I must say, that however different in symbolic expres
sion, Mr. Ellis s logical system seems to me identical in
principle with my own. The developments of the Com
binational Method, as described in the Educational Times
(June, July, and August, 1872), are substantially the same as
I had previously published in several papers and books.
Mr. Ellis also employs card diagrams of combinations
arranged upon the ledges of a black-board, which practi-
PREFACE. xvii
cally form the Logical Abacus, as described by me in
1869.
The only point in which I am conscious of having
received assistance from Mr. Ellis has regard to the
necessary presence of combinations and the significance
of their total disappearance as proving contradiction.
I may not have sufficiently insisted upon the importance
of this matter; but the fact is that so long ago as 1864
(see pp. 1 8 1, 192, of this book) I pointed out the complete
disappearance of ^a letter-term from the combinations as
the criterion of contradiction in the conditions governing
logical combinations, and the same principle is explicitly
stated in the Principles of Science (1874, vol. i. p. 133; new
edition, p. 116). In the latter part of this book I have
more fully developed the theory of the relation of pro
positions, often turning as it does upon this criterion of
contradiction. This theory will, I think, be found to be
the natural development of ideas stated in my earlier
essays; but I may have received some hints from
Mr. Ellis s writings. The above remarks apply only to
such portions of Mr. Ellis s Memoirs as treat of logical
combination and inference; other portions in which he
investigates sequence in space and time, probability,
etc., are not at all in question.
The Logical Index, although now printed for the first
time, has been in my possession since 1871 (see Principles
of Science, ist edition, vol. i. pp. 157, 162; new edition,
pp. 137, 141, etc.); but it is only by degrees that I have
appreciated the wonderful power which it gives over all
xviii PREFACE.
logical questions involving three terms only; and it is quite
recently that it has occurred to me how it might be printed
in the form of a compact and convenient table.
Mr. Venn has published in the Philosophical Magazine
for July 1880, a paper On the Diagrammatic and
Mechanical Representation of Propositions and Reason
ings. An article on Symbolic Reasoning by the same
author will also be found in Mind for the same month.
The text of this book having been completed and placed
in the printer s hands before Mr. Venn s ingenious papers
were published, it has not been possible to illustrate or
to criticise his views.
I may mention that M. Louis Liard, Professor of Philo
sophy at Bordeaux, who had previously explained and
criticised the substitutional view of Logic in the Revue
Philosophique (Mars, 1877, torn, iii., p. 277, etc.), has
since published a very good though concise account of the
principal recent logical writings in England, under the
title, Les Logiciens anglais contemporains (Paris : Germer
Bailliere, 1878).
These Studies consist in great part of logical Questions
and Problems gathered from many quarters. In the majority
of cases I have indicated by initial letters the source or
authorship of the questions when clearly known (see the
List of References on p. xxv) ; but I have not always
carried out this rule, and in not a few cases the questions
have been printed several times already, and are of
doubtful authorship. A large remaining fraction of the
questions and problems are new, and have been de-
PREFACE. xix
vised specially for this book. As shown by the author s
name appended, a few questions have been borrowed
from the work of the Very Rev. Daniel Bagot, Dean
of Dromore, entitled Explanatory Notes on the Principal
Chapters of Murray s Logic . . . with an Appendix of 3 3 7
Questions to Correspond. A few excellent illustrations
have also been drawn from a privately printed tract on
Logic by the late Sir J. H. Scourfield, M.P., his own
annotated copy having been kindly presented to me by
the author a few years before his death.
In forming this compilation I have been more than ever
struck by the fact that the larger part of logical difficulties
and sophisms do not turn upon questions of formal logic
but upon the relations which certain assertions bear to the
presumed or actual knowledge of the assertor and the
hearer. If the person X remarks that c All lawyers are
honourable men, it is one question what is the pure
logical force of this proposition, as measured by its
effect on the combinations of the terms concerned and
their negatives. It is quite another matter what X means
by it ; why he asserts it ; what he expects Y to understand
by it ; and what Y actually does take as the meaning
of X.
Under certain circumstances assertions convey a meaning
the direct opposite of what they convey at other times.
If a man is taken with a fit and the first medical man who
arrives says, You must not think of putting the man
under the pump/ the man will not be put under the
pump ; but if the identically same assertion is made about
xx PREFACE.
the centre of interest of an excited and angry mob, the
man goes to the pump. It is evident that there ought to
exist a science of applied deductive logic, partly corre
sponding to the ancient doctrines of rhetoric, in which
the popular force of arguments as distinguished from their
purely logical force should be carefully analysed. A few
questions and answers given in this book may perhaps
belong, properly speaking, to rhetorical logic (see pp. 119,
140-1, etc.), but I have not found it practicable to pursue
the subject in this book. It should be evident that a
thorough comprehension of the purely logical aspect of
assertions must precede any successful attempt to in
vestigate their rhetorical aspect. I may possibly at some
future time attack the problems of rhetorical logic.
A further question which forced itself upon my notice
was that of the practicability of including exercises in
Inductive Logic. As Mr. H. S. Foxwell suggested, in
ductive exercises and problems are even more needed
than those of a deductive character. But, on consideration
and trial, it seemed highly doubtful whether it would be
possible to throw questions of inductive logic into the
concise and definite form essential to a book of exercises.
I have given abundance of inverse combinational problems
which are really of an inductive character (see pp. 2528);
but exercises in the inductive methods of the physical
sciences, if practicable at all, would require a much greater
space, and a very different mode of treatment from that
which they could receive in this work. For the present,
at all events, I must content myself with referring readers
PREFACE. xxi
to the ample exposition of inductive methods contained
in the 3d, 4th, and 5th books of the Principles of
Science.
Some readers may perhaps be still inclined to object
to the Syllogism, and to deductive logic generally, that it
is comparatively worthless, because all new truths are
obtained by induction. This doctrine has prevailed with
many writers from the time of John Locke to that of John
Stuart Mill. But if I have proved in Chapters VI., VII.,
XL, XII., and in other parts of the Principles of Science,
that induction is the inverse operation of deduction, the
supreme importance of syllogistic and other deductive
reasoning is not so much restored as explained. In
reality the cavillers against the syllogism have never suc
ceeded in the slightest degree in weakening the hold
of the syllogism upon the human mind : it was against
the nature of things that they should succeed. Their
position was as sensible as that of a tutor who should
recommend his pupils to begin Mathematics with Compound
Division, but on no account to trouble themselves with the
obsolete formula of the Multiplication Table. In every
point of view, then, a thorough command of deductive
processes is the necessary starting-point for any attempt to
master more difficult and apparently more important pro
cesses of reasoning.
In the composition of the didactic parts of this book,
I have tried the experiment of throwing my remarks into
the form of answers to assumed, or in many cases actual,
examination questions. I cannot call to mind any book
xxii PREFACE.
in which this mode of treatment has been previously
adopted, but it seems to lend itself very readily to the
clear exposition of knotty points and difficulties. In spite
of much popular clamour against examinations, I maintain
that to give a clear, concise, and complete written answer to
a definite question or problem is not only the best exercise
of mind, but also the best test of ability and training,
which can be generally applied.
The Frontispiece contains rough facsimiles of ancient
logical diagrams which I copied from the fine MS. of
Aristotle s Organon in the Ambrosian Library at Milan
(L. 93, Superior). During a visit to Italy in 1874, I was
much surprised and interested by the multitudes of curious
diagrammatic exercises to be found in the logical MSS.
of the great public libraries of Italy. The abundance
of these diagrams shows that rudimentary logical exer
cises were very popular in the country where, and at the
time when, the dawn of modern science began to break. I
estimated that a single MS. in the Biblioteca Communale
at Perugia (Aristotelis de Interpretatiom cum Comment.
A, 55. Grsece. Chart. 1485) contained at least eight
hundred such diagrams. Those given in the frontis
piece are the most ancient which I could discover. The
MS. containing these (among others) is assigned in the
printed catalogue to the eleventh or twelfth century, but
the librarian was of opinion that it might belong to the
tenth century. The figure in the centre shows the Greek
original of the familiar Square of Logical Opposition, which
has survived to this day (see p. 31). The triangular and
PREFACE. xxiii
lunular figures represent respectively the syllogistic moods
Darapti, and (I believe) Datisi.
To the imperfect list of the most recent writings on
Symbolical Logic, given in this preface, I am enabled to
add at the last moment the important new memoir of
Professor C. S. Peirce on the Algebra of Logic, the first part
of which is printed in the American Journal of Mathematics,
vol. iii. (i5th September 1880). Professor Peirce adopts
the relation of inclusion, instead of that of equation, as the
basis of his system.
BRANCH HILL,
HAMPSTEAD HEATH, N.W.,
3^ October 1880.
NOTE TO SECOND EDITION.
THE present Edition has been printed from the Author s
own copy, in which he had marked the few corrections
and alterations which have now been made.
HARRIET A. JEVONS.
REFERENCE LIST
OF INITIAL LETTERS SHOWING THE AUTHORSHIP
OR SOURCE OF QUESTIONS AND PROBLEMS.
A = PROFESSOR ROBERT ADAMSON, Owens College,
Manchester.
B = PROFESSOR ALEXANDER BAIN, University of Aberdeen.
C = Cambridge University. Moral Science Tripos, or
College Examination Papers.
D = Dublin University.
E = Edinburgh University. PROFESSOR ERASER.
H = REV. JOHN HOPPUS, formerly Professor of Logic, etc.
in University College, London.
I = India Civil Service Examinations.
L = London University, Second B.A., Second B.Sc., M.A.,
M.D. and D.Sc. Examinations.
M = PROFESSOR THOMAS MOFFET, President and Professor
in Queen s College, Galway.
O = Oxford University.
P = PROFESSOR PARK, Queen s College, Belfast, and
Queen s former University.
R = PROFESSOR GROOM ROBERTSON, University College,
London.
W = WHATELY S Elements of Logic.
CONTENTS.
CHAPTER pAGE
I. THE DOCTRINE OF TERMS I
II. QUESTIONS AND EXERCISES RELATING TO TERMS . 9
III. KINDS OF PROPOSITIONS !8
!V. EXERCISES IN THE DISCRIMINATION OF PROPO
SITIONS 25
V. CONVERSION OF PROPOSITIONS, AND IMMEDIATE
INFERENCE 31
VI. EXERCISES ON PROPOSITIONS AND IMMEDIATE
INFERENCE ^
VII. DEFINITION AND DIVISION 64
VIII. SYLLOGISM t , l
!X. QUESTIONS AND EXERCISES ON THE SYLLOGISM . 94
X. TECHNICAL EXERCISES IN THE SYLLOGISM . .103
XI. .CUNYNGHAME S SYLLOGISTIC CARDS . . . 107
XII. FORMAL AND MATERIAL TRUTH AND FALSITY . m
XIII. EXERCISES REGARDING FORMAL AND MATERIAL
TRUTH AND FALSITY I22
XIV. PROPOSITIONS AND SYLLOGISMS IN INTENSION . 126
XV. QUESTIONS ON INTENSION ^
xxviii CONTENTS.
CHAPTER PAGE
XVI. HYPOTHETICAL, DILEMMATIC, AND OTHER KINDS
OF ARGUMENTS . . 137
XVII. EXERCISES IN HYPOTHETICAL ARGUMENTS . . 145
XVIII. THE QUANTIFICATION OF THE PREDICATE . .149
XIX. EXERCISES ON THE QUANTIFICATION OF THE
PREDICATE . 159
XX. EXAMPLES OF ARGUMENTS AND FALLACIES . .164
XXI. ELEMENTS OF EQUATIONAL LOGIC . 179
XXII. ON THE RELATIONS OF PROPOSITIONS INVOLVING
THREE OR MORE TERMS . . 223
XXIII. EXERCISES IN EQUATIONAL LOGIC . . 227
XXIV. THE MEASURE OF LOGICAL FORCE . . 249
XXV. INDUCTIVE OR INVERSE LOGICAL PROBLEMS . 252
XXVI. ELEMENTS OF NUMERICAL LOGIC . . 259
XXVII. PROBLEMS IN NUMERICAL LOGIC 276
XXVIII. THE LOGICAL INDEX 281
XXIX. MISCELLANEOUS QUESTIONS AND PROBLEMS . 290
STUDIES IN
DEDUCTIVE LOGIC.
CHAPTER I.
THE DOCTRINE OF TERMS.
INTRODUCTION.
1. IN accordance with custom, I begin this book of
logical studies with the treatment of Terms. Besides being
customary, this way of beginning is convenient, because
some difficulties which might otherwise be encountered in
the treatment of propositions and arguments are cleared
out of the way. But the continued study of logic convinces
me that this doctrine of terms is really a composite and for
the most part extra-logical body of doctrine. It is in fact
a survival, derived from the voluminous controversies of the
schoolmen.
2. The difficulties of metaphysics, of physics, of grammar,
and of logic itself, are entangled together in this part of
logical doctrine. Thus, if we take such a term as colour ;
and endeavour to decide upon its logical characters, we
should say that it is categorematic, because it can stand as
the subject of a proposition ; it is positive, because it im-
B
2 DOCTRINE OF TERMS. [CHAP.
plies the presence rather than the absence of qualities. But
is it abstract or concrete ? If concrete, it should be the
name of a thing, not of the attributes of a thing. Now
colour is certainly an attribute of gold or vermilion ; never
theless, colour has the attribute of being yellow or red or
blue. Thus I should say that yellowness is an attribute of
colour, and if so, colour is concrete compared with yellow
ness or blueness, while it is abstract compared with gold or
cobalt. If this view is right, abstractness becomes a question
of degree.
3. Again, a relative term is one which cannot be thought
except in relation to something else, the correlative. Thus
nephew cannot be thought but as the nephew of an uncle
or aunt; an instrument cannot be thought but as the
instrument to some end or operation. But the question
arises, Can anything be thought except as in relation to
something else ? What is the meaning of a table but as
that on which dinner is put ? What is a chair but the seat
of some person ? Every planet is related to the sun, and
the sun to the planets. Even meteoric stones moving
through empty space are related by gravity to the sun
attracting them. All is relative, both in nature and
philosophy.
4. As to the distinctions of general, singular, and proper
terms, connotative and non-connotative terms, etc., they
seem to me to be involved in complete confusion. I have
shown in the Elementary Lessons in Logic (pp. 4 J -44) tnat
Proper Names are certainly connotative. There would be
an impossible breach of continuity in supposing that, after
narrowing the extension of thing successively down to
animal, vertebrate, mammalian, man, Englishman, educated
at Cambridge, mathematician, great logician, and so forth,
thus increasing the intension all the time, the single re-
i.] DEFINITIONS AND EXAMPLES. 3
maining step of adding Augustus de Morgan could remove
all the connotation, instead of increasing it to the utmost
point. But however this and many other questions in the
doctrine of terms may be decided, it is quite clear in any
case that this part of logic is ill-suited for furnishing good
exercises in reasoning. This ground alone is sufficient to
excuse rny passing somewhat rapidly and perfunctorily over
the first part of logic, and going on at once to the subject
of Propositions which offers a wide field for useful exercises.
Accordingly, after giving brief definitions of the several
kinds of terms, a few answers to questions, and a fair
supply of unanswered questions and problems, I pass on to
the more satisfactory and prolific parts of logic.
DEFINITIONS AND EXAMPLES.
5. A general term is one which can be affirmed, in the
same sense, of any one of many (i.e. two or more) things.
Examples Building, .front-door, lake, steam-engine.
6. A singular term is one which can only be affirmed, in
the same sense, of one single thing.
Examples Queen Victoria, Cleopatra s Needle, the
Yellowstone Park.
7. A collective term is one which can be affirmed of
two or more things taken together, but which cannot be
affirmed of those things regarded separately or distributively.
Examples Regiment, century, pair of boots, baker s
dozen, book (a collection of sheets of paper).
4 DOCTRINE OF TERMS. [CHAP.
8. A concrete term is a term which stands for a thing.
Examples Stone, red thing, brute, man, table, book,
father, reason.
9. An abstract term is a term which stands for an
attribute of a thing.
Examples Stoniness, redness, brutality, humanity,
tabularity, paternity, rationality.
10. A connotative term is one which denotes a subject
and implies an attribute.
Examples Member of Parliament denotes Gladstone,
Sir Stafford Northcote, or any other individual
member of parliament, and implies that they can
sit in parliament.; bird denotes a hawk, or eagle,
or finch, or canary, and implies that they have all
the attributes of birds.
11. A non-connotative term is one which signifies an
attribute only, or (if such can be) a subject only.
Examples Whiteness denotes whiteness only, an
attribute without a subject. John Smith (according
to J. S. Mill, and some other logicians) denotes a
subject or person only, without implying attributes.
12. Concrete general names are always connotative.
Such also are all adjectives, without exception. Every
adjective is the name of a thing to which it is added, and
implies that the thing possesses qualities. Red is the name
of blood or of other red thing, and implies that it is red.
Redness is the abstract term, the name of the quality
redness.
i.] DEFINITIONS AND EXAMPLES. 5
13. A positive concrete term is applied to a thing in
respect of its possession of certain attributes ; a positive
abstract term denotes certain attributes.
Examples Useful, active, paper, rock; usefulness,
activity, rockiness.
14. A negative term is applied to a thing in respect of
the absence of certain attributes ; if abstract the term
denotes the absence of such attributes.
Examples Useless, inactive, not-paper ; uselessness,
inactivity.
15. An absolute term is the name of a thing regarded
per se, or without relation to anything else, if such there
can be.
Examples Air, book, space, water.
1 6. A relative term is the name of a thing regarded in
connection with some other thing.
Examples Father, ruler, subject, equal, cause, effect.
17. A categorematic term is one which can stand alone
as the subject of a proposition.
Examples Any noun substantive ; any adjective, any
phrase or any proposition used substantively.
1 8. A syncategorematic term is any word which can only
stand as the subject of a proposition in company with some
other words.
Examples Any preposition, conjunction, adjective used
adjectively.
19. Differences of opinion may arise concerning almost
every one of the definitions given above, and it would not
6 DOCTRINE OF TERMS. [CHAP.
be suitable to the purpose of this book to discuss the matter
further.
In every case, too, we ought before treating any terms
to ascertain clearly that there is no ambiguity about their
meanings. An ambiguous term is not one term, but two or
more terms confused together, and we should single out one
definite sense before we endeavour to assign the logical
characteristics. The ambiguity of terms has however been
sufficiently dwelt upon in the Elementary Lessons, Nos. iv.
and vi., and it need not be pursued here.
For the further study of the subject of terms the reader
is referred to the Elementary Lessons ; Mill s System of Logic,
book i., chapters i. and ii. ; Shedden s Logic, chapters i.
and ii. ; Levi Hedge s Logic, part ii., chapter i. ; Martineau,
Prospective Review, vol. xxix., pp. 133, etc. ; Hamilton s
Lectures on Logic, vol. iii., lectures viii. to xii. ; Woolley s
Introduction to Logic, part i., chapter i.
QUESTIONS AND ANSWERS.
20. Describe the logical characters of the follow
ing terms Equal, equation, equality, equalness,
inequality, and equalisation.
Equal is a noun-adjective ; concrete, as denoting equal
things ; connotative, as connoting the attribute of equality ;
general, positive, relative; and syncategorematic, because
it cannot as an adjective form the subject of a proposition.
Equation, noun-substantive, originally abstract, as mean
ing either equality, or the action of making equal. It is
now generally used by mathematicians to denote a pair of
i.] DEFINITIONS AND EXAMPLES. 7
quantities affirmed to be equal. It is thus concrete, general,
positive, perhaps absolute, and categorematic.
Inequality is a noun-substantive, abstract, singular, nega
tive, and categorematic.
Equalisation means the action of making equal, an attri
bute or circumstance of things, not a thing. It is thus
abstract, singular, positive, categorematic.
21. What are the logical characters of the terms,
drop of oil, oily, oiliness ?
A drop of oil being a concrete, finite thing, its name will
be concrete, general, positive, relative (as having dropped
from a mass of oil), collective as regards the particles of
oil, connotative as implying the qualities of oiliness, etc.,
and categorematic.
Oil is concrete, positive, collective, connotative and cate
gorematic, like drop of oil, and only differs in not admitting,
as regards any one kind of oil, of the plural. It is a case
of what I have proposed {Principles of Science, p. 28; ist ed.,
vol. i., p. 34) to call a substantial term, but which I find
that Burgersdyk, Heereboord, and the older logicians called
a totum homogeneum, the parts being of the same name
and nature with the whole. (Heereboord, Synopsis Logicae,
p. 83. See also Mind, vol. i. p. 210.)
Oily is a noun-adjective, and is concrete, general, positive,
connotative, as denoting oil and implying the attributes of
oiliness, doubtfully relative, syncategorematic, 1680.
Oiliness, noun-substantive, abstract, singular, positive,
categorematic.
Where distinctions are omitted, it may be understood that
they are regarded as inapplicable.
8 DOCTRINE OF TERMS. [CHAP. i.
22. Describe the logical characters of the terms
Related, relative, relation, relativeness, rela
tionship, relativity.
I have already dwelt, in the Elementary Lessons (p. 25),
on the prevalent abuse of the word relation, and other like
abstract terms. Nothing is more nearly impossible than to
reform the popular use of language ; but I will point out
once again that relation is properly the abstract name of the
connection or bearing of one thing to another, this being
an attribute of those things. The things in question are
properly said to be related, or to be relatives. Thus,
fathers, brothers, sisters, aunts, and cousins, are all relatives
not relations. Relationship is an abstract term signifying
the attribute of being related ; it was invented to replace
relation when this was wrongly used as a concrete term.
The relationship between a mother and her daughter is
simply the relation which exists between two such related
persons or relatives. Relativeness is an uncommon term
sometimes used to replace the abstract sense of relation,
where the case is not one of family relation. Relativity is
a further abstract term, probably due to Coleridge, and of
which the metaphysicians had better have the monopoly.
CHAPTER II.
QUESTIONS AND EXERCISES RELATING TO TERMS.
i. DESCRIBE the logical characters of the following terms,
classifying them according as they are
(a) Abstract or Concrete.
(b) General or Singular.
(c) Collective or Distributive.
(d) Positive or Negative.
(e) Absolute or Relative.
(/) Categorematic or Syncategorematic.
Prime Minister Biped
Institution Saturn
Copper Bismarck
Shameful Monarch
The London Library Unuseful
Collection The Times
School Board Paper
Deaf Augustus de Morgan
Equation John Jones
Innumerous John
Purpose Triangle
Function Musicalness
Cousin Board School
The Absolute Needlepoint
Black Representation
io DOCTRINE OF TERMS. [CHAP.
Europe Advocate
Injustice Being
Brace of partridges Whale
Dumbness Lawyer
Planetary System Time
Classification Manchester
2. In the case of the following terms distinguish with
special care between those which are abstract and those
which are concrete
Nature Animal Ethericity
Natural Animalism Scarce
Naturalness Animality Scarcity
Naturalism Animalcule Scarceness
Author Ether Truth
Authority Ethereal Trueness
Authorship Etherealness Verity
3. Investigate the ambiguity of any of the following
terms as regards their concrete or abstract character
Weight Science
Time Schism
Intention Space
Vibration Relation
4. Supply the abstract terms corresponding to the
following concrete terms
Wood Conduction
Stone Atmosphere
Conduct Alcohol
Witness Axiom
Equal Gas
Table Fire
Boy Socrates
ii.] QUESTIONS AND EXERCISES. n
5. In the case of such of the following terms as you
consider to be abstract, name the corresponding concrete
terms
Analysis Nation
Psychology Vacuity
Extension Realm
Production Folly
Socialism Evidence
6. Do abstract terms admit of being put in the plural
number ? Distinguish between the terms which are abstract
and concrete in the following list, and at the same time
indicate which can in your opinion be used in the plural :
colour, redness, weight, value, quinine, equation, heat,
warmth, hotness, solitude, whiteness, paper, space. [c.]
7. Investigate the logical characters and ambiguities of
the term form in all the following expressions : a religion
of forms ; the human form ; a form of thought ; a school
form ; a mere form ; a printer s form ; a form of govern
ment j form of prayer ; good form ; essential form.
8. What error is there in the following descriptions ?
Peerless syncategorematic, general, abstract, positive,
relative.
Bacon equivocal, concrete, general, substantial, positive,
relative.
Black categorematic, abstract, general, negative, abso-
1nf/=>
9. Analyse the following sentences as regards the logical
character of each term found in them, distinguishing
especially between such as are concrete or abstract,
collective or distributive, singular or general
12 DOCTRINE OF TERMS. [CHAP.
Logic is the science of the formal laws of thought.
Entre 1 homme et le monde il faut 1 humanite.
Art is universal in its influence ; so may it be in its
practice, if it proceed from a sincere heart and a quick
observation. In this case it may be the merest sketch, or
the most elaborate imitative finish.
10. Burton, in his Etmscan Bologna, p. 234, uses the
abstract term Etruscanidty. Is it possible in like manner
to make an abstract term corresponding to every concrete
one ? If so, supply abstracts for the following concretes-
Sir Isaac Newton. Royal Engineers.
Dictionary. Postal Telegraph.
11. What logical faults do you detect in the following
expressions ?
The standard authorship of modern times.
The three great nationalities of Western Europe.
The legal heir is not necessarily a man s nearest relation.
That unprincipled notoriety Pietro Aretino.
12. Coleridge, in a celebrated note to his Aids to Reflec
tion, thus defines an Idea : An Idea is the indifference of
the objectively real and the subjectively real : so, namely,
that if it be conceived as in the Subject, the idea is an
Object, and possesses objective truth ; but if in an Object,
it is then a Subject, and is necessarily thought of as exer
cising the powers of a Subject. Thus an Idea, conceived
as subsisting in an Object, becomes a Law : and a law
contemplated subjectively (in a mind) is an Idea?
Analyse the meanings of the terms Idea, Object, Subject,
Real, Truth, Law, etc., in the above passage, with respect
especially to their concreteness or abstractness. [L.]
IL] QUESTIONS AND EXERCISES. 13
13. Name the negative terms which correspond to the
following positive terms
Illumination Variable
White Famous
Certain Notorious
Constant Valid
Dying Plenty
14. Name the positive terms which correspond to the
following negative or apparently negative terms
Immensity Falsehood
Inestimable Unravelled
Disestablishment Infamous
Unpleasant Presuppositionless
Want Shameless
Unloosed Empty
Indifferent Intact
Headless Ignominious
15. In examining the following list of terms, distinguish,
as far as possible, between those which are really negative
in form and origin, and those which only simulate the
character of negatives
Annulled Undespairing
Disannulled Invalid
Antidote Headless
Infrequent Independence
Eclipse Individual
Undisproved Indolent
The Infinite Disagreeable
Impassioned Despairing
Immense Infant
Purposeless Deafness
i 4 DOCTRINE OF TERMS. [CHAP.
1 6. Can you find any examples of terms in the dictionary
which are true double negatives ? Paired, Impaired,
and Unimpaired, may perhaps be affirmed respectively
of two things which are equal, unequal, and not unequal.
Analyse the meaning of each of the following terms, and
show whether it is or is not a true double negative
Indefeasible Indefatigable
Uninvalided Uninjured
Undecomposable Undecipherable
Undefaceable Undeformed
Indestructible Indistinguishable
17. How are the denotation and connotation of a con
crete term related to the denotation of the corresponding
abstract term ?
1 8. Explain the difference of denotation and connotation
with reference to the terms Law, Legislator, Legality,
Crime. [L.]
19. Compare the connotation of the following sets of
terms
J Abbey j Caesar
( Westminister Abbey ( Roman
( Mineral c Road
< Oxide of iron < Means of communication
( Ore V Railway
20. Distinguish in the following list such terms as are
non-connotative, naming at the same time the logician
whose opinion on the subject you adopt
Virtue Socrates
Virtuous Barmouth
The mother of the Gracchi
Gladstone The Lord Chamberlain
ii.] QUESTIONS AND EXERCISES. 15
21. Form a list of twelve purely non-connotative names.
22. What is, if any, the connotation of these terms:
Charles the First ; Richelieu ; John Smith ; Santa Maria
Maggiore ?
23. Try to name half-a-dozen perfectly non- relative
names, and then inquire whether they really are non-relative.
What is the relation implied or involved in each of the
following terms ?
Metropolis County
Realm Alphabet
Capital city Sun
24. Show, by examples, that the division of Names into
general and singular does not coincide with the division
into abstract and concrete. [L.]
25. What kinds of words can stand as the subject of a
proposition, and what kinds are excluded ? [o.]
26. Distinguish between the distributive, collective, or
singular use of these Latin adjectives of quantity : omnis,
omnes, cunctus, cuncti, ullus, quidam, aliquis.
27. What is peculiar about the use of certain terms in
the following extracts ?
(1) Frenchmen, I ll be a Salisbury to you.
(2) His family pride was beyond that of a Talbot or a
Howard.
(3) In quo quisque artificio excelleret, is in suo genere
Roscius diceretur.
(4) When foe meets foe.
28. How does Logic deal with verbs, adverbs, and con
junctions ?
29. How many logical terms are there in the following
witty epigram ? Which and what are they ?
16 DOCTRINE OF TERMS. [CHAP.
What is mind ? No matter.
What is matter ? Never mind.
30. How many logical terms are there in each of the
following sentences? Ascertain exactly how many words
are employed in each such term.
(1) The Royal Albert Hall Choral Society s Concert is
held in the Albert Hall on the Kensington Gore Estate
purchased by the Royal Commissioners of the Great
Exhibition of 1851.
(2) A name is a word taken at pleasure to serve for a
mark which may raise in our mind a thought like to
some thought we had before, and which being pro
nounced to others, may be to them a sign of what
thought the speaker had before in his mind.
31. Words, says Hobbes, are insignificant (that is without
meaning), when men make a name of two names, whose
significations are contradictory and inconsistent : as this
name, an incorporeal body.
The following are a few instances of such apparently self-
inconsistent names, and the student is requested to add to
the list
(1) Corporation sole.
(2) Trigeminus.
(3) Manslaughter of a woman.
(4) An invalid contract.
(5) A breach of a necessary law of thought.
32. How would you explain the following apparent
absurdities ?
An Act of Parliament (1798-99) prohibited the importa
tion of * French lawns not made in Ireland.
ii.] QUESTIONS AND EXERCISES. 17
Ferguson (History of Architecture, Vol. II., p. 233) de
scribes a certain Moabite tower as a square Irish round
tower.
33. Are the following terms perfectly univocal or un
ambiguous, or can you point out any equivocation which is
possible in their use ?
Penny Lecture-Room
Charcoal Victoria Street
Aluminium Bible
Second Monday
34. Trace out and explain the ambiguities which affect
any of the following terms
Organ Stone March
Sole Corn Mood
Ear Diet Mean
Bowl Perch Force
Rock (stone) Bole Bowl
Rock (bird) Strait Straight
35. Draw out complete lists of all the words or expres
sions which have been developed out of the roots of the
following words (see Elementary Lessons in Logic, pp.
32 36, and Lesson VI.)
Post Logic
Section Faction
Final Function
Mission Decline
CHAPTER III.
KINDS OF PROPOSITIONS.
i. IN this chapter propositions will be described and
classed according to the ancient Aristotelian doctrine, in
which four principal forms of propositions were recognised,
thus tabularly stated :
Affirmative.
Negative.
Universal
Particular
Symbol = A
All X is Y
Symbol = E
No X is Y
Symbol = I
Some X is Y
Symbol = O
Some X is not Y
Singular propositions are to be classed as universal, and
indefinite propositions, in which no indication of quantity
occurs, must be interpreted at discretion as universal or
particular. The student is supposed to be familiar with
what the ordinary text-books say upon the subject.
I first give a series of Examples of propositions, with
brief comments upon their logical form and peculiarities.
A copious selection of exercises is then supplied in the
next chapter for the student to treat in like manner.
CHAP, in.] KINDS OF PROPOSITIONS. 19
EXAMPLES.
2. * Books are not absolutely dead things. O.
This proposition is indefinite or pre-indesignate, as
Hamilton would call it (Lectures on Logic, Vol. I. (III.),
p. 244); but, as we can hardly suppose Milton to have
thought that all books were living things, I take it to
mean some books are not, etc., that is to say, particular
negative.
3. The weather is cold. A.
The weather means the present state of the surrounding
atmosphere, and may be best described as a singular term,
which makes the assertion universal.
4. Not all the gallant efforts of the officers and escort
of the British Embassy at Cabul were able to save
them. E.
At first sight this seems to be a particular negative, like
Not all that glitters is gold ; but a little consideration
shows that gallant efforts is a collective whole, the efforts
being made in common, and therefore either successful or
unsuccessful as a whole. The meaning then is, The whole
of the gallant efforts, etc., were not able to saVe the men.
It is a universal negative.
5. * One bad general is better than two good ones. A.
This saying of Napoleon looks at first like a particular or
even a singular proposition ; but the one bad general
means not any definite one, but any one bad general
acting alone.
6. No non-metallic substance is now employed to make
money. E.
20 KINDS OF PROPOSITIONS. [CHAP.
The subject is a negative term, and the proposition might
be stated as All non-metallic substances are not any of
those employed to make money.
7. Multiplication is vexation.
If all multiplication is so, this is A ; there are certainly
other causes of vexation.
8. Wealth is not the highest good. E.
Affirmatively, wealth is one of the things which are not
the highest good.
9. Murder will out. A.
Like most proverbs, this is an unqualified universal
proposition ; its material truth may be doubted.
10. A little knowledge is a dangerous thing. A.
This looks like a particular affirmative, but is really
A, as meaning that any small collection of knowledge
is, etc.
11. All these claims upon my time overpower me. A.
Dr. Thomson points out (Outline, 5th ed., p. 131) that
all is here clearly collective.
12. The whole is greater than any of its parts. A.
Though apparently singular, this is really a general axiom,
meaning any whole is greater, etc.
13. No wolves run wild in Great Britain at the present
day. E.
1 4. Who seeks and will not take, when once tis offered,
shall never find it more. E.
This seems to be a compound proposition, but the sub
ject is, Any one who is seeking, but has not taken when
once it was offered.
in.] EXAMPLES. 21
15. The known planets are now more than a hundred
in number. A.
Clearly a collective singular affirmative proposition, and
therefore universal. Of course the planets separately could
not have the predicate here affirmed.
1 6. Figs come from Turkey. I.
Indesignate; that is to say, we cannot assume without
express statement that it is intended to say, All figs come
from Turkey.
17. Xanthippe was the wife of Socrates. A.
1 8. No one is free who is enslaved by his appe
tites. E.
19. Certain Greek philosophers were the founders of
logic. A.
Apparently I ; but if certain means a certain definite
group of men, each of whom was essential in his time,
the proposition becomes collective and singular, hence
universal.
20. Comets are subject to the law of gravitation. A.
Indefinite affirmative ; but in a matter of such universality
it may be interpreted as A.
21. Democracy ends in despotism. I.
Again indefinite ; but as referring to matter in which no
rigorous laws have been detected it should be interpreted
particularly.
22. Men at every period since the time of Aristotle have
studied logic. I.
Obviously particular as regards men.
22 KINDS OF PROPOSITIONS. [CHAP.
23. Few men know how little they know. O.
That is, Most men do not know, etc. Hence O.
24. Natura omnia dedit omnibus. A.
Singular affirmative, because natura is a singular term.
The assertion is one of Hobbes , and is thoroughly am
biguous as regards omnia and omnibus, which might be
capable either of collective or distributive meaning. No
doubt, however, the meaning is that Nature did not assign
anything to any particular person ; if so, both must be taken
collectively.
25. There are many cotton-spinners unemployed. I.
Really a kind of numerical assertion ; but if to be classed
at all, it must be I, many being only a part of all.
26. A^few Macedonians vanquished the vast army of
Darius. A.
Collective singular affirmative, because the few of course
acted together. It is a question whether the predicate is
not also singular.
27. True Faith and Reason are the soul s two eyes. A.
Collective singular.
28. A perfect man ought always to be busy conquering
himself. A.
All perfect men ought, etc.
29. A truly educated man knows something of every
thing and everything of something. A.
There seems to be two predicates, and hence a compound
sentence; but this is not the case, because the truly educated
man must know both.
in.] EXAMPLES. 23
30. Some comets revolve in hyperbolic orbits. I.
Particular affirmative as it stands.
31. The dividends are paid half-yearly. A.
* The dividends includes all so known.
32. Ov TO peya ev ecrrt, TO 8 ev /xeya. O and A.
This must mean that not all great things are good (O),
but that all good things are great (A). There are three
classes of things great and good ; great and not-good ;
not-great and not-good.
33. It is force alone which can produce a change of
motion. A.
It = what can produce, etc. The meaning is, Whatever
produces a change of motion is some kind of force ; but
there is no assertion that force = whatever produces, etc.
34. We have no king but Caesar.
As it stands, A ; but the meaning conveyed implies that
Caesar is our king ; Nobody who is not Caesar is our
king.
35. It is true that what is settled by custom though it
be not good, yet at least it is fit.
Complex; three propositions in all.
36. God did not make man, and leave it to Aristotle to
make him rational.
A simple and a singular negative proposition ; the not
applies to all that follows conjunctively, for of course Locke
could not have intended to assert that God did not make
man. * E.
24 KINDS OF PROPOSITIONS. [CHAP. in.
37. * Dublin is the only city in Europe, save Rome, which
has two cathedrals.
Compound sentence implying three propositions, namely
Dublin has two cathedrals. A.
Rome has two cathedrals. A.
All European cities, not being Dublin and not being
Rome, have not two cathedrals. E.
38. The affections are love, hatred, joy, sorrow, hope,
fear, and anger.
Really a disjunctive proposition. Affection is either love,
or hatred, or, etc. This implies that love is an affection,
hatred is an affection, etc.
CHAPTER IV.
EXERCISES IN THE DISCRIMINATION OF PROPOSITIONS.
i. EXAMINE each of the following propositions, and point
out in succession
(a) Which is the subject.
(b) Which is the predicate.
(c) Whether the proposition is affirmative or negative.
(d) Whether it appears to be universal or particular.
(e) Whether there is ambiguity or other peculiarity
in the proposition.
(1) All foraminifera are marine organisms.
(2) They never pardon who have done the wrong.
(3) Great is Diana of the Ephesians.
(4) No mammalia are parasites.
(5) Non progredi est regredi.
(6) Not every one can integrate a differential equation.
(7) All, all are gone, the old familiar faces.
(8) He that is not for us is against us.
(9) "Apurrov /xev vStop.
(10) Men mostly hate those whom they have injured,
(n) Old age necessarily brings decay.
(12) Nothing morally wrong is politically right.
(13) What I have written I have written.
(14) It is not good for man to be alone.
26 PROPOSITIONS. [CHAP.
(15) A certain man had a fig-tree,
(l 6) XaAeTra rot KaAa.
(17) There s something rotten in the state of Denmark.
(18) To be or not to be, that is the question.
(19) Ye are my disciples, if ye do all I have said unto you.
(20) Possunt qui posse videntur.
(21) There can be no effect without a cause.
(22) Rien n est beau que le vrai.
(23) Pauci laeta arva ten emus.
(24) All cannot receive this saying.
(25) Fain would I climb, but that I fear to fall.
(26) There s not a joy the world can give like that it
takes away.
(27) Not to know me argues thyself unknown.
(28) Two blacks won t make a white.
(29) Few men are free from vanity.
(30) He that fights and runs away may live to fight
another day.
(31) We are what we are.
(32) There is none good but one.
(33) Two straight lines cannot inclose space.
(34) Better late than never..
(35) Cruel laws increase crime.
(36) Omnes omnia bona dicere.
(37) Le genie n est qu une plus grande aptitude a la
patience.
(38) Whosoever is delighted in solitude is either a wild
beast or a god.
(39) Summum jus summa injuria.
(40) Non omnes moriemur inulti.
(41) Haud ignara mali miseris succurrere disco.
(42) Familiarity breeds contempt.
(43) Some politicians cannot read the signs of the times.
iv.J EXERCISES. 27
(44) Only the ignorant affect to despise knowledge.
(45) Recte ponitur; vere scire esse per causas scire.
(46) Only Captain Webb is able to swim across the
Channel.
(47) Some books are to be read only in parts.
(48) E pur si muove.
(49) Civilisation and Christianity are co-extensive.
(50) Some men are not incapable of telling falsehoods.
(51) Sunt nonnulli acuendis puerorum ingeniis non
inutiles lusus.
(52) All is not true that seems so.
(53) Me miserable.
(54) The Claimant, Arthur Orton, and Castro are in all
probability the same person.
(55) The three angles of a triangle are necessarily equal
to two right angles.
(56) Many rules of grammer overload the memory.
(57) Nullius exitium patitur natura videri.
(58) Summse artis est occultare artem.
(59) Wonderful are the results of science and industry
in recent years.
(60) Love is not love which alters when it alteration finds.
(6 1) A healthy nature may or may not be great ; but there
is no great nature that is not healthy.
(62) Quas dederis solas semper habebis opes.
(63) Quod volunt, id credunt homines.
(64) IIa(ra orap ov 8tKGua>$^o-eTai.
(65) Antiquitas seculi, juventus mundi.
(66) That would hang us, every mother s son.
(67) Men in great place are thrice servants.
(68) Justice is ever equal.
(69) A friend should bear a friend s infirmities.
(70) Men are not what they were.
28 PROPOSITIONS. [CHAP.
(71) The troops took one hour in passing the saluting
point.
(72) Nemo mortalium omnibus horis sapit.
(73) Fugaces labuntur anni.
(74) ATJTOS eyw et/jit.
(75) Communia sunt amicorum inter se omnia.
(76) Dictum sapienti sat est.
(77) The Romans conquered the Carthaginians.
(78) The fear of the Lord, that is wisdom.
(79) To live in hearts we leave below is not to die.
(80) Tis only noble to be good.
(81) Dum spiro spero.
2. In looking over the following list of propositions
distinguish between those which have a distributive and
those which have a collective subject.
(1) All the asteroids have been discovered during the
present century.
(2) All Albinos are pink-eyed people.
(3) The facts of aboriginal life seem to indicate that
dress is developed out of decorations.
(4) Non omnes omnia decent.
(5) Dirt and overcrowding are among the principal
causes of disease.
(6) Omnes apostoli sunt duodecim.
(7) Many artisans are unemployed.
(8) The side and diagonal of a square are incommen
surable.
(9) Omnis homo est animal.
(10) Nihil est ab omni parte beatum.
3. Ascertain exactly how many distinct assertions are
made in each of these sentences, and assign the logical
characters of the propositions.
iv.] EXERCISES. 29
(1) Tis not my profit that doth lead mine honour : mine
honour, it.
(2) True, tis a pity; pity tis, tis true.
(3) Hearts, tongues, figures, scribes, bards, poets, cannot
think, speak, cast, write, sing, number, ho ! his
love to Antony.
(4) A horse, a horse ! my kingdom for a horse.
(5) Istuc est sapere, non quod ante pedes modo est
videre : sed etiam ilia, quae futura sunt, prospicere.
(6) Virtue consists neither in excess nor defect of action,
but in a certain mean degree.
(7) The glories of our blood and state are shadows, not
substantial things.
(8) To gild refined gold, to paint the lily,
To throw a perfume on the violet,
To smooth the ice, or add another hue
Unto the rainbow, or with taper light
To seek the beauteous eye of heaven to garnish
Is wasteful and ridiculous excess.
(9) All places that the eye of heaven visits,
Are to a wise man ports and happy havens.
(TO) The age of chivalry is gone, and the glory of
Europe extinguished for ever,
(n) Poeta nascitur, non fit.
(12) Not all speech is enunciative, but only that in which
there is truth or falsehood.
(13) Devouring Famine, Plague, and War,
Each able to undo Mankind,
Death s servile emissaries are.
(14) Many are perfect in men s humours, that are not
greatly capable of the real part of business, which
is the constitution of one that hath studied men
more than books.
3 o PROPOSITIONS. [CHAP. iv.
(15) Vivre, ce n est pas respirer, c est agir.
(16) Justice is expediency, but it is expediency speaking
by general maxims, into which reason has concen
trated the experience of mankind.
(17) Men, wives, and children, stare, cry out, and run as
it were doomsday.
4. Distinguish so far as you can between the propositions
in the following list which are to you explicative and ampli-
ative. (See Elementary Lessons, pp. 68 69. Thomson s
Outline of the Necessary Laws of Thought, 81.)
(1) Homer wrote the Iliad and Odyssey.
(2) A parallelepiped is a solid figure having six faces, of
which every opposite two are parallel.
(3) The square on the hypothenuse of a right-angled
triangle is equal to the sum of the squares on the
sides containing the right angle.
(4) The swallow is a migratory bird.
(5) Axioms are self-evident truths.
5. Classify the following signs of logical quantity accord
ing as they are generally used to indicate universality,
affirmative or negative, or particularity, affirmative or
negative
Several, none, certain, few, ullus, nullus, nonnullus, not a
few, many, the whole, almost all, not all.
CHAPTER V.
CONVERSION OF PROPOSITIONS, AND IMMEDIATE INFERENCE.
i. THE student is referred to the Elementary Lessons in
Logic, or to other elementary text-books, for the common
rules of conversion and immediate inference, but for the
sake of easy reference, the ancient square of opposition is
given below.
A Contraries .... .E
C>
i Subcontraries . :o
All the relations of propositions and the methods of
inference applying to a single proposition will be found fully
exemplified and described in the following questions and
answers.
32 CONVERSION. [CHAP.
2. It appears to be indispensable, however, to endeavour
to introduce some fixed nomenclature for the relations of
propositions involving two terms. Professor Alexander
Bain has already made an innovation by using the name
obverse, and Professor Hirst, Professor Henrici and other
reformers of the teaching of geometry have begun to
use the terms converse and obverse in meanings incon
sistent with those attached to them in logical science
(Mind, 1876, p. 147). It seems needful, therefore, to state
in the most explicit way the nomenclature here proposed
to be adopted with the concurrence of Professor Robertson.
Taking as the original proposition all A are BJ the
following are what we may call the related propositions
INFERRIBLE.
Converse. Some B are A.
Obverse. No A are not B.
Contrapositive. No not B are A,
or, all not B are not A.
NON-INFERRIBLE.
Inverse. All B are A.
Reciprocal. All not A are not B.
It must be observed that the converse, obverse, and con-
trapositive are all true if the original proposition is true.
The same is not necessarily the case with the inverse and
reciprocal. These latter two names are adopted from the
excellent work of Delbceuf, Prolegomenes Philosophiques de
la Geometric, pp. 8891, at the suggestion of Professor
Groom Robertson. (Mind, 1876, p. 425.)
v.] QUESTIONS AND ANSWERS. 33
QUESTIONS AND ANSWERS.
3. Give all the logical opposites of the proposition,
All metals are conductors.
This is a universal affirmative proposition, having the
symbol A. By its logical opposites we mean the corre
sponding propositions in the forms E, I, and O, which
have the same subject and predicate, and are related to it
respectively as its contrary, contradictory, and subaltern, in
the way shown in the Logical Square (p. 31) and explained
in many Manuals. These opposite propositions may be
thus stated
Subaltern (I) Some metals are conductors.
Contradictory (O) Some metals are not conductors.
Contrary (E) No metals are conductors.
The first of these (I) may be inferred from the original ;
the other two (O and E), so far from being inferrible, are
inconsistent with its truth.
4. Given that a particular negative proposition
is true, is the following chain of inferences
correct ? O is true, A is false, I is false, and
therefore E is true. If so, the truth of O
involves the truth of E.
There is a false step in this argument ; for the falsity of
A does not involve the falsity of I. It may be (and is
materially false) that all men are dishonest ; but it never-
34 CONVERSION. [CHAP.
theless may remain true that some men are dishonest
Observe, then, that the falsity of A does not involve the
truth of I, nor does the truth of I involve the truth or
falsity of A. But the truth of A necessitates that of I.
As stated in the Elementary Lessons (p. 78), Of sub
alterns, the particular is true if the universal be true :
but the universal may or may not be true when the
particular is true.
5. How do you convert universal affirmative
propositions ?
They must be converted by limitation or per accidens^
as it is called, that is to say, while preserving the affirmative
quality, the quantity of the proposition must be limited from
universal to particular^ Thus A is converted into I, as
in the following more or less troublesome instances, the
Convertend standing first and the Converse second in each
pair of propositions :
f All organic substances contain carbon.
1 Some substances containing carbon are organic.
f Time for no man bides.
( Something biding for no man is time.
j The poor have few friends.
1 Some who have few friends are poor.
C A wise man maketh more opportunities than he finds.
< Some who make more opportunities than they find are
v wise men.
C They are ill discoverers who think there is no land, when
< they can see nothing but water.
\ Some ill discoverers think there is no land, etc.
v.] QUESTIONS AND ANSWERS. 35
( Great is Diana of the Ephesians.
I Some great being is Diana of the Ephesians.
( Warm-blooded animals are without exception air-breathers.
"\ Air-breathers are (with or without exception) warm-
v blooded animals.
6. How would you convert Brutus killed
Caesar?
The strictly logical converse is Some one who killed
Caesar was Brutus. For, though a man can only be killed
once, and Brutus is distinctly said to be the killer, yet in
formal logic we know nothing of the matter, and Caesar
might have been killed on other occasions by other persons.
An absurd illustration is purposely chosen in the hope that
it may assist to fix in the memory the all-important truth
that in logic we deal not with the matter.
7. How do you convert particular affirmative
propositions ?
To this kind of proposition simple conversion can be
applied ; that is to say, the converse will preserve both the
quantity and the quality of the convertend. In other
words, I when converted gives another proposition in I ;
thus either of the following pairs is the simple converse of
the other :
(" Some dogs are ferocious animals.
(. Some ferocious animals are dogs.
f Some men have not courage to appear as good as they are.
s Some, who have not courage to appear as good as they
v are, are men.
f Some animals are amphibious.
\ Some amphibious beings are animals.
36 CONVERSION. [CHAP.
8. How do you convert universal negative pro
positions ?
These also are converted simply, giving another universal
negative proposition. E gives E. The reason is that both
the terms of E are distributed ; a universal negative asserts
complete separation between the whole of the subject and
the whole of the predicate. No man is a tailed animal
asserts that not any one man is found anywhere in the class
of tailed animals. Hence it follows evidently that no one
being belonging to the class of tailed animals is found in
the class of men, which result we assert in the simple con
verse proposition, no tailed animal is a man. Further
examples of the same mode of conversion are given below.
f No virtue is ultimately injurious.
I No ultimately injurious thing is a virtue.
C No wise man runs into heedless danger.
\ No one who runs into heedless danger is a wise man.
/ People will not look forward to posterity who never look
J backward to their ancestors.
I People never look backward to their ancestors who will
v not look forward to posterity.
Whatever is insentient is not an animal.
Whatever is an animal is not insentient.
9. How do you convert particular negative
propositions ?
Difficulty arises about this question, because the first
rule of conversion tells us to preserve the quality of the
proposition ; the converse accordingly should be negative.
But a negative proposition always distributes its predicate,
v.] QUESTIONS AND ANSWERS. 37
because a thing excluded from a class must be excluded
from every part of the class. Now the subject of O being
particular and indefinite, it cannot stand as a distributed
predicate. It is still possible to say with material truth,
some men are not soldiers ; but converted this gives the
absurd result, all soldiers are not men ; or, no soldiers
are men. Even if we insert the mark of quantity some
before the predicate, and say, all soldiers are not some
men, we must remember that some is perfectly indefinite,
and may include all. The question will be more fully dis
cussed further on, but, so far as I can see, the particular
negative proposition, so long as it remains negative and
indefinite in meaning, is incapable of conversion. This fact
constitutes a blot in the ancient logic.
Nevertheless the proposition O is capable of giving a
converse result when we change it into the equivalent affirma
tive proposition. If some men are excluded from the
class * soldiers/ they are necessarily included in the class
non-soldiers, or, some men are non-soldiers. This is a
proposition in I, and by simple conversion, as already de
scribed, gives a converse also in I, some non-soldiers
are men. As further examples take
( Some dicotyledons have not reticulate leaves.
1 Some plants with non-reticulate leaves are dicotyledons.
J Some crystals are not symmetrical.
( Some unsymmetrical things are crystals.
J All men have not faith.
( Some who have not faith are men.
/ Not every one that saith unto me, Lord, Lord, shall enter
\ into the Kingdom of Heaven.
^ Some who shall not enter into the Kingdom of Heaven
\ say unto me, Lord, Lord.
38 CONVERSION. [CHAP.
10. How do you convert singular propositions ?
Singular propositions, being those which have a singular
term as subject, may be divided into two classes, according
as the predicate is a singular or a general term. (See
Karslake, 1851, vol. i., p. 54.) The former will always be
converted simply, one single thing being identified with the
same under another name, as in Queen Victoria is the
Duchess of Lancaster, converted into the Duchess of
Lancaster is Queen Victoria. Simple conversion will also
apply if the predicate be a general term, provided that the
proposition be negative so as to distribute this term. Thus,
St. Albans is not a great city becomes * no great city is
St. Albans. But if the predicate be general and undis
tributed, as in an affirmative singular proposition, then we
must convert per accidens, and limit the new subject to some
or even one significate of the general term. Examples of
each case follow :
( The better part of valour is discretion.
( Discretion is the better part of valour.
J Time is the greatest innovator.
1 The greatest innovator is time.
f London is the greatest of all cities.
(. The greatest of all cities is London.
{London is not a beautiful city.
No beautiful city is London.
( Le style est 1 homme meme.
1 L homme meme est le style.
( All the allied troops fought courageously.
1 Some who fought courageously were the allied troops.
v.] QUESTIONS AND ANSWERS. 39
( Mercy but murders, pardoning those that kill.
{ Something which murders is mercy, pardoning those that
V kill.
(Not all the figures that Babbage s calculating machine
could run up, would stand against the general heart.
Something which would not stand against the general heart
is all the figures (collectively) that Babbage s machine
could run up.
II. Show how to convert the propositions
(1) All mathematical works are not difficult
(2) All equilateral triangles are equiangular.
(3) No triangle has one side equal to the
other two.
The first proposition, as it stands, is ambiguous, for it
looks like the universal negative, no mathematical works
are difficult. But, according to custom, we may interpret
it as meaning that not all mathematical works are difficult,
or some mathematical works are not difficult, a proposition
in the form O. This cannot be converted simply, as already
explained (p. 36), because we must preserve the negative
quality, and all (or some) difficult things are not mathe
matical works being negative would distribute its predicate
mathematical works. We can, however, make O into I,
some mathematical works are not-difficult things, and we
can convert this simply into some not-difficult things are
mathematical works.
Proposition (2), as it stands, is in A, and can only be
logically converted by limitation into some equiangular
triangles are equilateral. Geometrically it could easily be
shown that the inverse proposition all equiangular triangles
are equilateral, is also true ; but we must of course not
40 CONVERSION. [CHAP.
allow knowledge of the matter in question to influence us
in logical deduction, and the inverse proposition cannot
be inferred from the original.
Number (3) is a universal negative, and must be converted
simply into * Nothing having one side equal to the other two
is a triangle; but there is something paradoxical about
this result, which the student is recommended to investigate.
12. Convert Life every man holds dear.
This is an example given in the Elementary Lessons,
(p. 304). Students have variously converted it into
Life is held dear by every man.
Some life is held dear, etc.
No man holds death dear (!)
and so forth. But it ought surely to be easy to see that the
grammatical object is transposed, life being the object of
holds dear. The statement is that every man holds life
dear, and is explicitly a universal affirmative proposition,
to be converted by limitation into some who hold life dear
are men.
13. Convert the proposition It rains.
What is it that rains ? What is it ? Surely the environ
ment, or more exactly the atmosphere. The proposition
then means the atmosphere is letting rain fall. The con
verse will therefore be something which is letting rain
fall is the atmosphere. But in this and many other cases
the Aristotelian process of conversion by limitation gives a
meaningless if not absurd result.
14. Convert the proposition He jests at scars
who never felt a wound.
This is the 8th example on p. 304 of the Elementary
v.] QUESTIONS AND ANSWERS. 41
Lessons, and has elicited from time to time some amazing
efforts at conversion, such as
Some jests at scars are made by one who never felt
a wound.
Scars are jested at by him who, etc.
Some scars jest at him who never felt a wound, (sic.)
Some scars are jests to one who, etc.
The subject of the proposition is of course he who never
felt a wound, and the proposition asserts that he thus
described jests at scars. As there is no limitation of
quantity we may take the subject as universal ; and, although
there is negation within the subject, the copula is affirmative,
and the proposition is in the form A. It is thus converted
by limitation into some who jest at scars are persons who
have never felt a wound.
15. Convert the proposition P struck Q.
To this simple question I have got answers that, since P
is distributed, and Q undistributed, we must convert by
limitation, getting some Q struck P ; or by contraposition
some not- Q struck not-/ . 7 Such blunders and nonsense
arise from failing to notice that struck is not a simple
logical copula. There is, of course, a relation between P
and <2; but as regards P, the proposition simply asserts
that P is a person who struck Q, possibly not the only
one. Hence the converse by limitation is some person
who struck Q is P.
Not a few examinees would at once convert P struck
Q into <2 struck /*, but this, although very likely to
happen materially, is not logically necessary.
16. What is the obverse of the proposition All
metals are elements ?
42 IMMEDIATE INFERENCE. [CHAP.
The obverse is a new term introduced by Professor
Alexander Bain, and its meaning is thus described by him
in his Deductive Logic, pp. 109, no. In affirming one
thing, we must be prepared to deny the opposite : " the
road is level," " it is not inclined," are not two facts, but
the same fact from its other side. This process is named
OBVERSION. He proceeds to point out that each of the
four propositional forms, A, I, E, O, admits of an obverse.
Every X is Y becomes no X is not- K Some X is V
becomes some X is not not- K No X is Y becomes all
X is not-K Some X is not Y 1 becomes some X is not-
y. Accordingly the obverse of the proposition above will
be No metals are not elements.
Professor Bain goes on to describe what he calls Material
Obversion, justified only on an examination of the matter
of the proposition. Thus from warmth is agreeable, he
infers, after examination of the subject-matter, that cold is
disagreeable. If knowledge is good, ignorance is bad. I
feel sure, however, that this mixing up of so-called material
obversion with formal obversion is likely to confuse people
altogether. Indeed, Mr. Bain is himself confused, for he
cites, I don t like a curving road, because I like a straight
one as a childish reason, being no reason at all, but the
same fact in obverse. Now, if there is any relation at all
between these two propositions, it is certainly a case of
material obversion ; but in reality they do not express the
same fact at all. The formal obverse of I like a straight
road, is I am not one who does not like a straight road.
We might perhaps infer, I do not dislike a straight road ;
but there is clearly no reference to curved roads at all.
While accepting the new term obversion in the sense of
formal obversion, I must add that students have begun to
use it with the utmost laxity, confusing the obverse with the
v.] QUESTIONS AND ANSWERS. 43
converse, the contrapositive, etc. To prevent logical nomen
clature from falling into complete chaos, it seems to be
indispensable to choose convenient names for the simpler
relations of prepositional forms, as attempted above (p. 32),
and to adhere to them inflexibly.
17. What is conversion by contraposition ? Give
the contrapositive of All birds are bipeds.
There is nothing which I have found so difficult in
teaching logic as to get the student to comprehend and
remember this process of contraposition ; particular attention
is therefore requested to the above question.
Having a proposition in A, we get its contrapositive by
taking the negative of its predicate, and affirming of this as
a subject the negative of the original subject. Thus, if all
Xs are Fs, we take all not-Fs as a new subject, and affirm
of them that they are all not-Xs, getting the proposition all
not- Fs are not-JTs, which is either A or E, according as
we do or do not join the negative particle to the predicate X.
Accordingly the contrapositive of the proposition all birds
are bipeds will be all that are not bipeds are not birds.
It is one thing to obtain the contrapositive, another thing
to see that it may be inferred from the premise. The late
Professor De Morgan used to hold that the act of inference
is a self-evident one, and needs no analysis ; but the process
may certainly be analysed. Thus we may obvert the pre
mise All Xs are Fs, obtaining No- J^s are not Fs, which
is a proposition in E, and then convert simply into * No
not-Fs are Xs, also in E, or else All not-Fs are not Xs.
The contrapositive, then, is the converse of the obverse.
We may also prove the truth of the contrapositive
indirectly; for what is not- F must be either X or not-^Y";
44 IMMEDIATE INFERENCE. [CHAP.
but if it be X it is by the premise also K, so that the same
thing would be at the same time not- Y and also J 7 , which
is impossible. It follows that we must affirm of not-K
the other alternative, not-X. (See Chapter XXI. below;
also Principles of Science, pp. 83, 84 ; first ed., vol. i., pp.
97, 98.)
18. Give the converse of the contrapositive of
the proposition All vegetable substances are
organic.
As learnt from the last question, the contrapositive is
All not-organic substances are not vegetable substances.
We may take this to be equivalent to No inorganic sub
stances are vegetable substances (E), the simple converse
of which is No vegetable substances are inorganic sub
stances, the obverse of the premise. But, if we treat the
contrapositive as a universal affirmative proposition, thus,
All inorganic substances are non-vegetable substances/
we must convert by limitation, getting Some non-vegetable
substances are inorganic, which is the subaltern of the
obverse, and cannot by any process of inference lead us
back to the original. Conversion by limitation is easily
seen to be a faulty process which always occasions a loss
of logical force.
As we shall afterwards observe, this kind of conversion
introduces a new term, namely the indeterminate adjective
* some, so that the inference is not really confined to the
terms of the original premise. Although we may not be
able to dispense entirely with the word, owing to its employ
ment in ordinary discourse, we shall ultimately eliminate it
from pure formal logic, and relegate it to the branch of
numerical logic.
v.] QUESTIONS AND ANSWERS. 45
19. Take the following proposition, all water
contains air ; convert it by contraposition :
change the result into an affirmative pro
position, and convert.
To show the need of more careful logical training than
has hitherto been common, even in the great Universities,
I give a few specimens of answers which I received to the
above question. The contrapositive of the proposition was
variously stated, as
All air does not contain all water.
All air is not contained in water.
All not-air is not a thing contained by not-water.
Some air is not contained in water.
Some not-air contains no water.
All not-air contains water.
The logicians who drew these inferences then proceeded
by simple conversion to get such results as the following :
Some water is not without some air.
No water contains not some air.
No water contains no air.
One too clever student inferred that All or every vacuum
is a void of water, which he converted, simply indeed, into
Every void of water is a vacuum !
An examiner in logic is sometimes forced to believe that
there is a void in the brains of an examinee ; but the
absence of any sufficient training in logical work is more
often the cause of the lamentable results shown above. In
any case it seems impossible to agree with De Morgan that
contraposition is a self-evident process.
46 IMMEDIATE INFERENCE. [CHAP.
These absurd answers are mainly due to the failure to
observe that in the proposition All water contains air, the
two words contains air, form the grammatical predicate,
comprehending both the logical predicate and the logical
copula. Logically then the proposition is All water is
containing air, or All water is what contains air. The
contrapositive then is All that does not contain air is not
water. Uniting the negative particle to the predicate
* water, and converting by limitation, we obtain Some
not-water is what does not contain air.
20. Describe the logical relations, if any, between
each of the following propositions and each
other
(1) All organic substances contain carbon.
(2) There are no inorganic substances which do
not contain carbon.
(3) Some inorganic substances do not contain
carbon.
(4) Some substances not containing carbon are
organic.
Of these, (i) is a universal affirmative, the contrapositive
of which is All substances not containing carbon are inor
ganic substances. Hence the converse by limitation of this
contrapositive is, Some inorganic substances are substances
not containing carbon, equivalent to (3).
Proposition (2) is the obverse of All inorganic substances
contain carbon, which is the contradictory of (3).
To obtain (4) we must take the contrary of (i), that is,
no organic substances contain carbon, express it in the
affirmative form, All organic substances are substances
not containing carbon, and then convert it by limitation,
v.] QUESTIONS AND ANSWERS. 47
21. Take any proposition suitable for the pur
pose, convert it by contraposition, convert it
again simpliciter, change the result into an
affirmative proposition, and show that you
may regain the original proposition. [c.]
The most suitable kind of proposition for the purpose
will be a universal affirmative, such as
(1) All birds are bipeds.
The contrapositive may be stated in the form of E.
(2) No not-bipeds are birds.
Which is converted simpliciter into E, the obverse of (i).
(3) No birds are not-bipeds.
When thrown into the affirmative form by a second
obversion, the last becomes
(4) All birds are not-not-bipeds.
As double negation destroys itself, this is equal to (i).
Notice that the obverse of the obverse is the original.
22. Give the converse of the contradictory of the
proposition, There are no coins which are not
made of metal.
The premise is stated in a complex form with double
negation; it means - No coins are not made of metal,
which is the obverse of All coins are made of metal (A).
The contradictory, as shown in the square of opposition
(p. 31) is a proposition in O, namely, Some coins are
not made of metal, which can be converted only by nega
tion, that is, by joining the negative particle to the predicate,
thus: Some coins are not -made -of- metal, whence by
simple conversion Some things not-made-of-metal are
coins the answer required.
48 IMMEDIATE INFERENCE. [CHAP.
23. (i) All crystals are solid.
(2) Some solids are not crystals.
(3) Some not-crystals are not solids.
(4) No crystals are not-solids.
(5) Some solids are crystals.
(6) Some not-solids are not crystals.
(7) All solids are crystals.
Assign the logical relation, if any, between each
of these propositions and the first of them.
Proposition (i) is a universal affirmative (A); its simple
obverse is (4); its converse by limitation is (5); the sub-
contrary of this converse is (2). In order to obtain (6)
we must take the contrapositive of (i), namely, All not-
solids are not crystals/ the subaltern of which is (6); and
converting (6) by negation we get (3). Again, (7) is the
inverse, but is not inferrible from (i). We may further say
that (4) can be inferred from (i), and is exactly equivalent
in logical force to it ; (5) and (6) can be inferred, but are
not equivalent to the original; (2) cannot be inferred from
(i), but is not inconsistent with its truth.
24. What information about the term not-^4 can
we derive from the premise * All As are s ?
This question, though apparently a very simple one, does
not admit of a very simple answer ; it is important in a
theoretical point of view. It may be said on the one hand,
that as the proposition only affirms of all As that they are
JBs, this tells us nothing about things excluded from the
class A. Thus what is not-A may be B, or it may not be
JB, without any interference from the premise. This is quite
true. About Not- A universally we may infer nothing.
But, on the other hand, if we convert the proposition all
As are s by contraposition (p. 43), we get all not-^s
v.] QUESTIONS AND ANSWERS. 49
are not As. Uniting the negative particle to the predicate,
we have All not-^s are not-^s, whence, by limited
conversion, we infer some not-^s are not-^s. In this
result we must interpret some as meaning, one at least, it
may be more or even all. We shall recur to this question
in a subsequent chapter.
25. Assuming that no organic beings are devoid
of carbon, what can we thence infer respectively
about beings which are not organic, and things
which are not devoid of carbon ?
The premise No organic beings are devoid of carbon
is a universal negative proposition, and does not directly
give information about beings which are not organic, and
beings which are not devoid of carbon. But, if we join the
negative particle to the predicate, we get All organic
beings are not-devoid-of-carbon, whence, by limited con
version, Some things not devoid of carbon are organic,
which answers the second part of the question.
Again, converting by contraposition, we learn that All
things not-not-devoid of carbon are not organic beings ; in
other words, * All things devoid of carbon are not organic
beings, a result which may be obtained perhaps more clearly
by converting the original premise simply, thus, No things
devoid of carbon are organic beings, or All things devoid
of carbon are not organic beings. Conversion by limitation
then yields Some things not organic beings are devoid of
carbon, which is the answer to the first part of the question.
This result is the same as that obtained in the last question,
and the same remarks apply.
26. What information about the term Solid Body
can we derive from the proposition, No bodies
which are not solids are crystals ?
E
50 IMMEDIATE INFERENCE. [CHAP.
This question differs from the last only in being put in a
more involved form. The premise when more simply
stated becomes All not solids are not crystals, the contra-
positive of All crystals are solids, and limited conversion
gives Some solids are crystals.
27. Nihil potest placere, quod non decet. Con
vert this proposition, (i) simply, (2) by con
traposition ; and show by what logical processes
we can pass back from the contrapositive to
the original. [c.]
This premise (from Quinctilian, c. xi. 65) equals, Nihil quod
non decet, potest placere; nothing which is unbecoming can
please. Being a universal negative, E, it can be converted
simply into Nothing which can please is unbecoming.
In order to apply contraposition, we must put the pre
mise into the form of A, thus All unbecoming things are
unpleasing things, the contrapositive of which is All not
unpleasing things are not unbecoming things, which having
a double negative in each term equals All pleasing things
are becoming. We can regain the original premise by
applying contraposition to this last result.
28. Convert, and give some immediate inferences
from the following: Nothing is harmless that
is mistaken for a virtue.
The predicate of this proposition is clearly harmless/
and that is mistaken for a virtue is a relative clause
describing the subject. The proposition is then Nothing
mistaken for a virtue is harmless/ (E), converted simply
into another proposition in E, Nothing harmless is mis
taken for a virtue.
Applying obversion to the original proposition we get
v.] QUESTIONS AND ANSWERS. 51
All that is mistaken for a virtue is not-harmless, or is
harmful. By immediate inference by complex conception,
we infer All foolish conduct mistaken for virtue is harmful
foolish conduct. (Concerning inference by complex con
ception, see Thomson s Outline, 88, and Elementary Lessons,
P- 87.)
29. Because every Prime Minister is a man, can
we infer that every good Prime Minister is a
good man ?
The process of immediate inference by added deter
minants, as described by Dr. Thomson, allows us to join an
adjective or determining mark to both terms of an affirmative
judgment, narrowing both terms, but to the same extent.
Of course, however, it must be the same determining mark
in each case, and if an adjective be ambiguous it is not
logically the same adjective in its several meanings. Now
good applied to a Prime Minister means that he is an able,
active, upright minister, but probably very different from
men who are good in other ranks of life. A good man
means one who is good in the ordinary business and
domestic relations of life. Thus the inference is erroneous.
(See Elementary Lessons, p. 86.) It will afterwards be shown
that when the proposition is fully expressed no such failure
of inference can occur. Strictly speaking the premise is
Prime Minister = Prime Minister, Man ;
and it follows inevitably that
Good, Prime Minister = Good, Prime Minister, Man.
30. Euler employed two overlapping circles to
represent a particular proposition. Can you
raise any objection to the accuracy of such a
diagram ?
52 IMMEDIATE INFERENCE. [CHAP.
Such circles have been employed in a great number of
logical works. In my Elementary Lessons (p. 75) the
particular proposition some metals are not brittle, is
represented by the following figure :
FIG. i.
It does not seem to have been sufficiently noticed that
though such a diagram correctly shows the exclusion of a
part of the class metals from any part, that is all parts, of
the class brittle substance, it indicates at the same time that
another part of the class metals is included among brittle
substances. Thus the diagram corresponds to the two pro
positions I and O, instead of showing either apart from the
other. Now, it has been fully explained that O is consistent
with the truth of E ; so that when we say * some metals are
not brittle, it may be that no metals are brittle, which is
contradictory to I, some metals are brittle. The diagram
should not prejudice this question, and it would therefore
be best to remove the part of the circle bounding metals
which falls within the circle of brittle substances, or else to
have a broken line, as in Fig. 2.
FIG. 2.
v.] QUESTIONS AND ANSWERS. 53
In the same way the proposition I, for instance, Some
crystals are opaque/ would be represented by a broken circle
included within a complete circle, in the manner shown
either in Fig. 3 or Fig. 4.
/ 3
/ 2
( I
o
FIG. 3. FIG. 4.
31. What is the logical force of the following
sentence from Sidgwick s Methods of Ethics :
A materialist will naturally be a determinist;
a determinist need not be a materialist ?
Taking naturally to give a universal force to the first
proposition, it becomes All materialists are determinists.
The second proposition informs us that a determinist need
not be a materialist, that is to say, at the least, some
determinists are not materialists. This proposition is the
sub-contrary of the converse of the first, and is the con
tradictory of all determinists are materialists. The second
proposition, then, prevents us from supposing materialists
and determinists to be two co-extensive terms. We learn
that there are persons called materialists who are all found
among determinists; hence some called determinists are
found among materialists ; other determinists, however, are
not among materialists, and as to those who are not deter
minists, they cannot be materialists. The first proposition
would be technically described as A, and the second as O
the contradictory of the inverse of the first.
54 IMMEDIATE INFERENCE. [CHAP.
32. All equilateral triangles are equiangular.
May we thence infer that triangles having
unequal angles have unequal sides, and vice
versa f
The proposition asserts that all equal-sided triangles have
equal angles ; hence we may by contraposition infer that
triangles which have not equal angles cannot have equal
sides. But as the proposition stands, we are not justified in
reading it reciprocally (see p. 32), and inferring that triangles
which have not equal sides have not equal angles. This is
true as a matter of geometrical science, but it is the contra-
positive of another proposition, namely, the inverse all
equiangular triangles are equilateral/ the truth of which
must be separately proved.
33. Can we ever convert a proposition of the
form all As are >s into one of the form all
s are As ?
Certainly we cannot infer that all s are As because all
As are JBs. As a general rule the predicate of the con-
vertend B will be a wider term than the subject A, so that
the inverse could not be inferred. Professor Henrici
(Elementary Geometry, Congruent Figures, p. 14), for in
stance, describes space as a three-way-spread, but we cannot
convert simply, and say that every three-way-spread is space.
It nevertheless happens not uncommonly that the original
proposition is really intended to mean all As are all JBs,
which can then be simply converted. Thus if space be
defined as a three-way-spread of points, we can convert into
every three-way-spread of points is space. Such definitions
are of the form of proposition afterwards described by the
symbol U (chapter xviii.), and considerable care is requisite
v.] QUESTIONS AND ANSWERS. 55
in discriminating between the propositions A and U.
J. S. Mill has pointed to the simple conversion of a
universal affirmative proposition as a very common form
of error (System of Logic, book v., chapter vi., section 2).
It cannot be too often repeated that the reciprocal and
inverse propositions as described on p. 32, cannot be
inferred from an original of the form A.
34. In what cases does predication involve real
existence ? Show that in some processes of
conversion assumptions as to the existence of
classes in nature have to be made ; and illus
trate by examining whether any such assump
tions, and if so what, are involved in the
inference that if all 5 is P, therefore some
not-S is not P.
The above question must have been asked under some
misapprehension. The inferences of formal logic have
nothing whatever to do with real existence ; that is, occur
rence under the conditions of time and space. No doubt,
if all S is P, it follows that, in order to avoid logical con
tradiction, some not-6 1 must be admitted to be not P. For
instance, if All heathen gods are described in writings
more than 1000 years old, it follows that Some things
which are not heathen gods are not described in writings
more than 1000 years old. This involves no assertion
of real existence, nor could such an inference ever be
drawn, unless, indeed, the original proposition itself asserted
existence in time and space. This subject is pursued in
a subsequent chapter.
CHAPTER VI.
EXERCISES ON PROPOSITIONS AND IMMEDIATE INFERENCE.
1 . EXAMINE the following pairs of propositions, and decide
which pairs contain consistent propositions, such that if the
first of the pair be true the second may be true ; and vice
versa^ if the second be true, the first may be true. Give
the technical name of the logical relation, if any, between
the two propositions of each pair.
/ \ J Some metals are useful.
I All metals are useful.
. ^ ( No metals are useless.
i Some useful things are not metals.
Some useless things are metals.
All useful things are metals.
Some metals are useful.
No metals are useless.
All metals are useful.
Some useless things are not metals.
2. Draw all the immediate inferences you can from the
proposition Knowledge is power.
3. Give the converse of the contrapositive of the pro
position All organic substances contain carbon.
CHAP, vi.] EXERCISES ON PROPOSITIONS. 57
4. Give all the logical opposites of intuta qua indecora,
1 Unsafe are all things unbecoming.
5. What information about the term * solid body can
we derive from the proposition, No bodies which are not
solids are crystals ?
6. Only British subjects are native born Englishmen.
What precisely does this proposition tell us about the four
terms British subject.
Not-British-subject.
Native born Englishmen.
Not-native-born-Englishmen ?
7. Describe the logical relation between each of the
four following propositions, and each of the other three :
(1) All substances possess gravity which are material.
(2) No substances which possess gravity are immaterial.
(3) Some substances which are immaterial do not possess
gravity.
(4) Some substances which do not possess gravity are
immaterial.
8. State the nature and technical name of the logical
process by which we get each of the following propositions
from the preceding one :
All men are mortal.
No men are immortal.
No immortals are men.
None but mortals are men.
All not-mortals are rot men.
No men are not-mortals.
All men are mortals.
9. What are the subaltern propositions corresponding
to the following universal propositions ?
58 IMMEDIATE INFERENCE. [CHAP.
(1) Every effect follows from a cause.
(2) No one is admitted without payment.
(3) All trespassers will be prosecuted with the utmost
rigour of the law.
(4) Nemo me impune lacessit.
10. Give the obverse, converse, inverse, and reciprocal
of each of the following propositions :
(1) All mammalia are vertebrate animals.
(2) Sir Rowland Hill is dead.
(3) That which is a merit in an author is often a fault in
a statesman.
(4) Whatever is necessary exists.
(5) In veritate victoria.
11. Give the contrary, contradictory, subaltern, converse,
obverse, inverse, contrapositive, and reciprocal propositions
corresponding to each of the following propositions :
(1) All B.A. s of the University of London have passed
three examinations.
(2) All men are sometimes thoughtless.
(3) Uneasy lies the head which wears a crown.
(4) The whole is greater than any of its parts.
(5) None but solid bodies are crystals.
(6) He who has been bitten by a serpent is afraid of a
rope.
(7) He who tries to say that which has never been said
before him, will probably say that which will never
be repeated after him.
1 2. Give as many equivalent logical expressions as you
can for the propositions
(i) If the treasury was not full, the tax-gatherers were
to blame.
vi.] EXERCISES. 59
(2) Through any three points not in a straight line a
circle may be described.
(3) It is false to say that only the virtuous prosper in
life. [R.]
13. What logical relations are there between the following
propositions ?
(1) All elementary substances are undecomposable.
(2) There are no compounds which are not decom
posable.
(3) Some compounds are not decomposable.
(4) No undecomposable substances are compounds. [E.]
14. From the proposition Perfect happiness is im
possible can we infer that Imperfect happiness is
possible ?
15. Is it the same thing to affirm the falsity of the pro
position Some birds are predatory, and to affirm the truth
of the proposition Some birds are not predatory ?
1 6. Explain the statement that in the case of subcontrary
propositions, truth may follow from falseness, but falseness
cannot follow from truth.
17. Give in succession (i) the obverse, (2) the converse,
(3) the subaltern, (4) the contrary, (5) the contradictory,
(6) the contrapositive of the proposition All wise acts are
honest acts.
1 8. Concerning the same proposition answer the following
questions :
(1) How is its converse related to its subaltern?
(2) How is its converse related to the converse of its
subaltern ?
(3) How is its subaltern related to its contradictory?
[BAGOT.]
60 IMMEDIATE INFERENCE. [CHAP.
19. What is the converse of the contrary of the con
tradictory of the proposition Some crystals are cubes ?
How is it related to the original proposition ?
20. What is the converse of the converse of No men
are ten feet high ?
21. Name the logical process by which we pass from
each of the following propositions to the succeeding
one :
(1) All metals are elements.
(2) No metals are non-elements.
(3) No non-elements are metals.
(4) All non-elements are not metals.
(5) All metals are elements.
(6) Some elements are metals.
(7) Some metals are elements.
22. (i) None but a logical author is a truly scientific
author. Taking this proposition as a premise, examine
the following propositions, and decide which of them can
be inferred from the premise.
(2) A truly scientific author is no author who is not
logical.
(3) Some truly scientific authors are not any authors
who are not logical.
(4) A not truly scientific author is not a logical author.
(5) Those who are not truly scientific authors cannot
be logical.
(6) All logical authors are truly scientific.
(7) No truly scientific author is an illogical author.
(8) All not illogical authors are truly scientific.
(9) No illogical author is a truly scientific author.
(10) No one is a truly scientific author who is not a logical
author,
(n) Some logical authors are not truly scientific authors.
vi.] EXERCISES. 61
Give, as far as possible, the technical name of the logical
relation between each of the above propositions and each
other.
23. Some small sects are said to have no discreditable
members, because they do not receive such, and extrude all
who begin to verge upon the character. Point out how
this statement illustrates logical conversion.
24. Can we logically infer that because heat expands
bodies, therefore cold contracts them ?
25. Does it follow that because every city contains a
cathedral, therefore the creation of a city involves the
creation of a cathedral, or the creation of a cathedral in
volves the creation of a city ?
26. All English Dukes are members of the House of
Lords. Does it follow by immediate inference by complex
conception that the creation of an English Duke is the
creation of a member of the House of Lords ?
27. Give every possible converse of the following pro
positions
(1) Two straight lines cannot enclose space.
(2) All trade-winds depend on heat.
(3) Some students do not fail in anything. [M.]
28. Give the logical opposites, converse and contra-
positive, of Euclid s (so-called) twelfth axiom
If a straight line meet two straight lines, so as to make
the interior angles on the same side of it taken together less
than two right angles, those straight lines being continually
produced shall at length meet upon that side on which are
the angles which are less than two right angles.
29. How is the above proposition related to this other :
If a straight line fall upon two parallel straight lines, it
62 IMMEDIATE INFERENCE. [CHAP.
makes the two interior angles upon the same side together
equal to two right angles ? [R.]
30. From Some members of Parliament are all the
ministers (Elementary Lessons, p. 325, No. 3 [4]), can we
infer that some place -seeking prejudiced and incapable
members of Parliament are all the place-seeking prejudiced
and incapable ministers ?
31. Is it perfectly logical to argue that because two sub-
contrary propositions may both be true at the same time,
therefore their contradictories, which are contrary to each
other, may both be false ?
32. Is it perfectly logical to argue thus? If contrary
propositions are both false, their respective contradictories,
which are sub -contraries to each other, are both true.
Now as this result is possible, it is therefore possible that
the contraries may both be false.
33. What is the logical relation, if any, between the
two assertions in Proverbs, chap. xi. i, A false balance
is abomination to the Lord: but a just weight is his
delight ?
34. Examine the verses of Proverbs, chap. x. to xv.
and assign the relation between the two opposed assertions
which make nearly all the verses.
35. What is the nature of the step from anger is a short
madness to madness is a long passion ? [R.]
36. The angles at the base of an isosceles triangle are
equal. What can be inferred from this proposition by
obversion, conversion, and contraposition, without any
appeal to geometrical proof?
37. From the assertion The improbable is not impos
sible, what can we learn, if anything, about (i) the
possible, (2) the probable, (3) the not-improbable, (4) the
impossible, (5) the not-impossible ?
vi.] EXERCISES. 63
38. How would a logician express the relations between
the following statements of four interlocutors ?
(1) None but traitors would do so base a deed.
(2) And not all traitors.
(3) Some would.
(4) No ; not even traitors.
[College Moral Science Examination, Cambridge.]
39. What difficulties or absurdities do you meet in con
verting the following propositions ?
(1) Some books are dictionaries.
(2) No triangle has one side equal to the sum of the
other two.
(3) Every one is the best judge of his own interests.
(4) A few men are both scientific discoverers and men
of business.
(5) Whatever is, is right.
(6) Some men are wise in their own conceit.
(7) The eye sees not itself,
But by reflection, by some other things.
CHAPTER VII.
DEFINITION AND DIVISION.
i. ALMOST all text-books of Deductive Logic give rules
for judging of the correctness of definitions, and for
dividing up notions into subaltern genera and species. On
attempting, however, to treat these parts of logic in the
manner of this work, it has come home to me very strongly
that they are beyond the sphere of Formal Logic, and
involve the matter of thought. In form there is nothing
peculiar to a definition ; in fact the very same proposition
may be a definition to one person and a theorem to another.
It is open to us for instance to define the number 9 as
9 = 3 x 3; or, 9 = 8 + 1; or, 9 = 7 + 2, etc.; but having selected
at will any one of these equations as a definition, the other
equations follow as theorems. The perplexity in which the
theory of parallel lines is involved partly arises from the
fact that there is choice of definitions, some mathematicians
choosing one way and some the other. It is quite apparent,
too, that the same proposition may aftord different know
ledge to different people. For instance, John Herschel
was the only son of William Herschel would serve as a
definition of John Herschel to any one who knew only
William Herschel, and of William Herschel to one who
only knew John. To one who knew both it might be a
CHAP, vii.] EXERCISES. 65
theorem. Similar remarks might be made concerning the
distinction between ampliative and explicative propositions.
2. These in addition to other considerations convince me
that any attempt to treat definition as a part of Formal
Logic must be theoretically unsound and practically un
satisfactory. The case is somewhat similar with Logical
Division, which, so far as it belongs to Formal Logic, can
be nothing more than that method of Dichotomous Division
fully developed in the later chapters on Equational Logic.
Anything more than this must involve material knowledge,
and should be treated in a different work and in a different
manner. On these grounds I have decided not to attempt
any explication of Definition and Division here, but to con
fine this chapter to a collection of questions, such as are to
be commonly found in examination papers. The student
may be referred for the current doctrines to the Elementary
Lessons ; Nos. XII. and XIII.; Fowlers Deductive Logic,
Chapters VII. and VIII. ; Duncarfs Logic, Chapter VI. etc.
3. Examine the following definitions
(1) Conversion is the changing of terms in a proposi
tion.
(2) Opposed propositions are those which differ in
quantity and quality.
(3) Contradictory opposition is the opposition of con
tradictories. [R.]
4. Define any of the following terms, notions, or classes
of objects
Gravitation Franchise Communism
Consistency Imagination Honour
Library Honesty Club
Vegetable Revenge Dictionary
66 DEFINITION AND DIVISION. [CHAP.
Diet Syllogism Conservative
Hypochondriac Racehorse University
Merit Success Specie.
5. Criticise the following definitions
(1) A square is a four -sided figure of which the sides
are all equal and the angles all right angles.
(2) A member of the solar system is anything over which
the sun has continued influence.
(3) La vie est le contraire de la mort.
(4) A lemma is a proposition which is only used as
subservient to the proof of another proposition.
(5) An archdeacon is one who exercises archidiaconal
functions.
(6) Life is the definite combination of heterogeneous
changes, both simultaneous and successive, in
correspondence with external coexistences and
sequences.
(7) A gentleman is a man having no visible means of
subsistence. [ORTON.]
(8) Equal bodies are those whereof every one can fill
the place of every other. [HOBBES.]
6. Examine the definitions
(1) Tin is a metal lighter than gold.
(2) Vice is the opposite of virtue.
(3) Paper is a substance made of rags.
(4) Cheese is a caseous preparation of milk.
(5) Rust is the red desquamation of old iron.
(6) A transcendental function is any function which is
not an algebraic function.
(7) A right-angled triangle is a triangle containing one
right angle, and of which the containing sides
are or are not equal.
vii.] EXERCISES. 67
(8) An organ is any part of an animal or plant appro
priated to a distinct function.
(9) A man is a self-knowing animal.
(10) Knowledge is that on which somebody else can be
examined. [ROLLESTON.]
( 1 1 ) An animal is a sentient organised being.
(12) A triangle is a three-sided figure having its angles
together equal to two right angles.
(13) A man is one who may be the Prince of Tran
sylvania. [HOBBES.]
7. In what respects are the following definitions, or some
of them, defective ?
(1) Logic is a guide to correct reasoning.
(2) Logic is the art of expressing thoughts in correct
language.
(3) Logic is a mental science.
(4) Logic is the science of the regulative laws of human
thought.
8. Does the eleventh chapter of the Hebrews, or any
part of it, contain a correct logical definition of Faith ?
9. Give examples of indefinable words, and explain why
words may be indefinable.
10. Give the Proximate Genera for the following
species
Man Plant Monarchy
Triangle Hound Science.
1 1. Define by genus and differentia the following terms ;
and name a proprium and an accident in each case :
Island Parallelogram
Bank Bill of exchange
Dictionary Tree.
68 DEFINITION AND DIVISION. [CHAP.
1 2. What are the genus, species, difference, property, and
accident of Examination ? [D.]
13. Distinguish specific attribute, property, and accident,
using the term Circle as an example. [B.]
14. Is it possible to define the terms gold, coal, legal
nuisance, civilisation, Cleopatra s needle?
15. Define the term boat, and then point out how many
of the following things the definition includes : Bark,
ferry-boat, floating fire-engine, pontoon, wherry, canoe.
1 6. Classify the following objects under one or other of
the heads, cash, bills, specie :
Cheque, promissory note, shilling, money, token -coin,
bank-note, I.O.U., paper -money, sovereign, Scotch bank
note. (See Money and the Mechanism of Exchange (Inter
national Scientific Series), p. 248. Section on the Definition
of Money.)
17. Distinguish Logical from Physical Division and
Definition. [o.]
1 8. Can anything admit of more than one defini
tion ? [o.]
19. Distinguish precisely between the definition and the
description of a class.
20. Explain the difficulties which arise concerning the
definition of parallel straight lines, and criticise the following
suggested definitions :
(1) Lines which are in every part equidistant.
(2) Lines of similar direction.
(3) Lines which being in the same plane and indefinitely
prolonged never meet.
vii.] EXERCISES. 69
21. Examine the following definitions :
(1) Man is a bundle of habits.
(2) Law is common sense.
(3) Reverence is the feeling which accompanies the
recognition of superiority or worth in others.
(4) Hunger is the product of man s reflection on the
necessity of food. [p.]
22. Which of the following are logical divisions, and
which are not ?
(1) Man into civilised and uncivilised.
(2) The world into Asia, America, Europe, Africa,
Australia.
(3) Grammar into syntax and prosody.
(4) War into civil and aggressive.
(5) Syllogisms into those which are logical and illogical.
(6) Sequences of phenomena into casual and causal.
(7) Energy into potential and visible.
(8) Geometrical figures into plane and tri-dimensional.
(9) Allegiance is either natural and perpetual, or local
and temporary.
23. Divide the term Inference, so as to include the
various species usually discussed by logicians. [E.]
24. The following were the classes of persons which were
in 1868 qualified to vote in one or other of the United
States of America : Male citizen, male inhabitant, every
man, white male citizen, white freeman, male person, white
male adult, free white male citizen, free white man.
Form a scheme of logical division which shall have a
place for each of the above classes.
25. Divide logically University, colour, chair, science,
70 DEFINITION AND DIVISION. [CHAP. vn.
religion, species, art, church, undergraduate, virtue, states
man, [o.]
26. Form a scheme of division of sciences to include the
species Deductive, experimental, concrete, descriptive,
rational, abstract, inductive, explanatory, empirical.
27. Apply the rules of logical division to the following
instances, correcting what is wrong, and supplying what is
deficient :
(1) Discursive thought may be divided into the Term,
Judgment and Syllogism.
(2) Notions are Concrete, Singular, and Universal.
(3) Propositions are Affirmative, Negative, and Universal.
28. To what extent are the rules of division, usually
given in logical treatises, repudiated by the classifications
adopted in the Natural Sciences ? [L.]
29. When is a division inadequate? When indistinct?
When a cross division ? And when not arranged according
to proximate parts ? [MORELL.]
30. Give an accurate scheme of logical division in which
the following things shall find places: Name; Part of
Speech ; Term ; Vox logica ; Verb ; Noun, Adjective ;
Syncategorematic term ; Word.
CHAPTER VIII.
SYLLOGISM.
i. MEDIATE Inference, or Syllogism, forms the principal
part of Deductive Logic, and offers a wide scope for useful
exercises. I give, in the first place, a brief epitome of the
syllogistic rules and forms ; I then exemplify them abun
dantly by question and answer ; lastly, I supply chapters
full of the largest and most varied collection of syllogistic
questions and problems which has ever been published.
Some of the more perplexing questions, involving the dis
tinction of formal and material falsity of syllogisms and
their premises, are treated apart in the succeeding
chapter (xii.).
RULES OF THE SYLLOGISM.
1 i ) Every syllogism has three and only three terms.
These terms are called the major term, the minor term,
and the middle term.
(2) Every syllogism contains three and only three proposi
tions.
These propositions are called respectively the major
premise, the minor premise, and the conclusion.
(3) The middle term must be distributed once at least.
(4) No term must be distributed in the conclusion which
was not distributed in one of the premises.
72 SYLLOGISM. [CHAP.
(5) From negative premises nothing can be inferred.
(6) If one premise be negative, the conclusion must be nega
tive ; and vice versa, to prove a negative conclusion one of the
premises must be negative.
From the above rules may be deduced two subordinate
rules, which it will nevertheless be convenient to state at
once.
(7) From two particular premises no conclusion can be
drawn.
(8) If one premise be particular, the conclusion must be
particular.
FIGURES OF THE SYLLOGISM.
S = minor term. M = middle term. P = major term.
First Figure. Second Figure. Third Figure. Fourth Figure.
M...P P...M M...P P...M
Minor S ... M S ... M M ... S M ... S
Premise.
Minor
Premise.
Conclusion. S...P S...P S...-T b...x
MOODS OF THE SYLLOGISM.
The following is a compact table of the valid moods of
the syllogism, the numerals showing the figures in which
each group of propositions makes a valid syllogism :
AAA AAI AEE All AGO
i. 3-4- 2.4. 1.3. 2.
EAE EAO EIO IAI OAO
1.2. 3.4- 1.2.3.4. 3.4. 3.
MNEMONIC VERSES.
Barbara, Celarent, Darii, Ferioque, rmoris ;
Cesare, Camestres, Festino, Baroko, secundae ;
viii.] QUESTIONS AND ANSWERS. 73
Tertia, Darapti, Disamis, Datisi, Felapton,
Bokarclo, Ferison, habet ; Quarts insuper addit,
Bramantip, Camenes, Dimaris, Fesapo, Fresison.
Certain letters in the above lines indicate the way in
which the moods of the second, third, and fourth figures
may be reduced to the first figure, as follows
s directs you to convert simply the proposition denoted
by the preceding vowel.
p directs you to convert the proposition per accident, or
by limitation.
m, for muta, directs you to transpose the premises.
k denotes that the mood can only be reduced per
impossibile.
The initial consonant of each mood in the three last
figures corresponds with the initial of the mood of the first
figure to which it is reducible.
QUESTIONS AND ANSWERS.
2. State the figure and mood to which the sub
joined argument belongs :
Iron is not a compound substance; for iron
is a metal, and no metals are compounds.
The conjunction * for shows that the proposition pre
ceding is the conclusion a universal negative. The term
* metal must be the middle term, because it does not ap
pear in the conclusion. The major term being compound
substance, the major premise must be * no metals are com
pound substances, (E) and the other premise iron is a
metal must be the minor. The latter is a universal affirm-
74 SYLLOGISM. [CHAP.
ative ; for though no mark of quantity is prefixed to iron,
it states a chemical truth concerning iron in general, and may
fairly be interpreted universally (A). The argument belongs
to the mood EAE in the first figure, or Celarent, thus :
E No metals are compound substances.
A (all) Iron is a metal.
E Iron is not a compound substance.
3. Examine the following argument ; throw it
into a syllogistic form, and bring out the
figure and mood :
It cannot be true that all repression is mis
chievous, if government is repressive and yet
is sometimes beneficial. [B.]
The conclusion is stated in the form of a denial of the
universal affirmative all repression is mischievous ; hence
the contradictory of this, or some repression is not mis
chievous is the real conclusion. The middle term is
government/ which does not appear in the conclusion.
In looking for the major term, we do not find mischievous
in the premises, but only its opposite term beneficial.
We must assume, then, that we are intended to take bene
ficial as equivalent to not-mischievous, otherwise there
would be a fallacy of four terms. To be brief, then, the
syllogism takes this form
Some government is not mischievous.
All government is repressive (or repression).
Therefore, some repression is not mischievous. It is a
valid syllogism in the third figure, and mood OAO, or
Bokardo.
VIII.} QUESTIONS AND ANSWERS. 75
4. In what figures is the mood AEE valid ?
In the first figure we have
All M is P.
No S is M.
No S is P.
The negative conclusion distributes the major term P,
which is undistributed in the major premise ; hence Illicit
Process of the Major Term.
In the second figure we have
All P is M.
No S is M.
No S is P.
The major term is now properly distributed in the major
premise, and the middle term being also distributed once,
in the minor negative premise, the syllogism is valid in
Camestres.
The reader may show that in the third figure we have
again Illicit Process of the Major, and in the fourth figure
a valid syllogism Camenes.
5. What rules of the syllogism are broken by
arguments in the pseudo-moods, OAE, and
OIE?
The answer cannot be better given than in the words of
Solly (Syllabus of Logic, p. 86). In the mood OAE the
predicate is distributed in the major premise, and the
subject in the minor premise, and both subject and predi
cate in the conclusion. Hence it follows that either some
term must be distributed in the conclusion which was not
distributed in the premises, or else the middle term cannot
76 SYLLOGISM. [CHAP.
be distributed in either premise. We cannot, therefore,
determine at once which form the fallacy will take, but may
be quite certain that there must be either an illicit process
of major or minor, or else an undistributed middle.
Again, in the mood OIE, both subject and predicate
are distributed in the conclusion, whereas no term is distri
buted in the minor premise, and it therefore follows that
there must be an illicit process of the minor. It is also
evident that the middle term cannot be distributed in
the minor premise, and that if it is distributed in the
major premise the major term must be undistributed, and
consequently there must be a fallacy either of undistributed
middle or illicit major.
6. None but whites are civilised ; the ancient
Germans were whites : therefore they were
civilised. [w.]
This appears at first sight to be in Barbara, the terms
standing apparently in the order of the first figure. But the
major premise does not assert that all whites are civilised ;
it only asserts that none but whites are so, and this is equiva
lent to the contrapositive of the proposition
All civilised are whites.
Joining to this the minor premise
The ancient Germans were whites,
we see that the argument is in the second figure, with two
affirmative premises, so that the middle term is undistri
buted in both cases, producing Fallacy of Undistributed
Middle. There is also a difference of tense between the
two premises which might perhaps invalidate an argument ;
but this point need not be further noticed here.
vin.] QUESTIONS AND ANSWERS. 77
7. None but civilised people are whites ; the
Gauls were whites : therefore they were
civilised. [w.]
At first sight this seems to be in the second figure, and
invalid; but converting the major premise by contraposition,
as in the last example, we get a valid syllogism in Barbara
thus, All whites are civilised; the Gauls were whites, etc.
8. All books of literature are subject to error ;
and they are all of man s invention ; hence
all things of man s invention are subject to
error. [H.]
This may seem at the first reading to be correct reason
ing, especially as the conclusion is materially true ; but
there is fallacy of Illicit Process of the Minor Term. The
argument is in the pseudo-mood AAA of the third figure,
and the conclusion should be * some things of man s invention
are subject to error.
9. He who is content with what he has is truly
rich ; a covetous man is not content with what
he has ; no covetous man, therefore, is truly
rich.
The middle term is content with what he has, and since
this term appears as subject of the major premise and
predicate of the minor, the argument is in the first figure in
the pseudo-mood AEE. There is Illicit Process of the
Major Term, because the conclusion E distributes its
predicate and the major premise A does not.
The conclusion may be true in matter but does not follow
from the premises. We could only make the argument good
7 8 SYLLOGISM. [CHAP.
by taking as major premise, All the truly rich are content
with what they have. This would give a valid syllogism in
Camestres, but the original premise, if converted, only
yields Some truly rich are content with what they have.
10. Protection from punishment is plainly due to
the innocent ; therefore, as you maintain that
this person ought not to be punished, it appears
that you are convinced of his innocence, [w.]
The above is equivalent to
The innocent are not to be punished ;
This person is not to be punished ;
Therefore, this person is innocent.
Put in this form there is an obvious fallacy of Negative
Premises ; but we can also express the premises in an
affirmative form as follows :
The innocent ought to be exempt from punishment ;
This person ought to be exempt from punishment.
In this case it is apparent that the middle term ought to be
exempt from punishment is undistributed in both the
premises, against Rule 3 of the syllogism.
11. He that is of God heareth my words : ye
therefore hear them not, because ye are not
of God. [w.]
In the usual order :
He that is of God heareth my words ;
Ye are not of God ;
. . Ye do not hear my words.
The propositions are AEE in the first figure, and involve
viii.] QUESTIONS AND ANSWERS. 79
the Fallacy of Illicit Process of the Major Term. Hear
my words is distributed as the predicate of the negative
conclusion, but is undistributed as the predicate of the
affirmative major premise. The argument would become
valid, however, if we were allowed to quantify this predicate
universally, and assume the meaning to be
He that is of God = he who heareth my words.
12. Any books conveying important truths with
out error deserve attention ; but as such books
are few, it is plain that few books do deserve
attention.
Carefully distinguish the truth and fallacy in
this argument.
A good example suggested by Whately s No. 13 ; it may
be thus put :
Any books conveying, etc., deserve attention ;
Few books do convey, etc. ;
. . Few books do deserve attention.
This is in the first figure, and, if we interpret the con
clusion to mean that ^ A few books do deserve attention,
that is to say, affirmatively only, without implying that the
rest do not, it is valid in the mood Darii. But usually
(Elementary Lessons, p. 67), we interpret/^/ negatively;
indeed, in the example itself this is the plain meaning, such
books are few, implying that all but this few do not convey
important truths without error. This makes the minor pre
mise into O, Most books do not, etc., and the argument
consisting of AOO in the first figure is a case of Illicit
Process of the Major Term.
8o SYLLOGISM. [CHAP.
13. That man is independent of the caprices of
Fortune, who places his chief happiness in
moral and intellectual excellence : A true
philosopher is independent of the caprices of
Fortune : therefore a true philosopher is one
who places his chief happiness in moral and
intellectual excellence. [w.]
A case of the fallacy of Undistributed Middle, the middle
term independent of the caprices of Fortune, being pre
dicated affirmatively both of one who places his chief
happiness, etc., and of a true philosopher. The pseudo-
mood is, therefore, A A A in the second figure. The fallacy
is none the better because the conclusion may be considered
true in matter. If the premise had begun Only that man
is independent, etc., we might have put the argument into
a valid syllogism, Barbara.
14. It is an intensely cold climate that is sufficient
to freeze quicksilver ; and as the climate of
Siberia does this it is intensely cold.
At the first glance this looks like a case of Undistributed
Middle ; but we soon see that the major premise is really
Any climate sufficiently cold to freeze quicksilver is an
intensely cold climate. The argument is thus valid in
Barbara.
15. No one who lives with another on terms of
confidence is justified, on any pretence, in
killing him : Brutus lived on terms of confi
dence with Caesar : therefore he was not justi
fied, on the pretence he pleaded, in killing
him. [w.]
viii.] QUESTIONS AND ANSWERS. 81
This is valid in Barbara, the major term being * justified on
any pretence, etc., the middle term, one who lives, etc.,
and the minor term, Brutus.
The conclusion, however, is obviously weakened, or is
less general than it might have been. We might conclude
that Brutus was not justified in killing Caesar on any pretence.
It is only inferred that he was not justified in killing him on
the pretence he pleaded, which is of course included in any
pretence.
16. Inquire into the validity of the following
argument : Whatever substance is properly
called by the name Coal consists of a carbon
aceous substance found in the earth ; now, as
this specimen consists of a carbonaceous sub
stance, and was found in the earth, therefore it
is properly called Coal. [L.]
The above argument is evidently a case of Undistri
buted Middle, because we infer that this specimen is
properly called Coal on the ground of two universal affirma
tive propositions, both of which have the same predicate
consisting of a carbonaceous substance found in the earth.
The pseudo-mood then is AAA in the second figure.
Though entirely failing in a demonstrative point of view,
it is another question whether the specimen may not be
believed to be coal, on analogical or inductive inference.
17. Give any remarks which occur to you con
cerning the following : Nerve power does not
seem to be identical with electricity; for it is
found that when a nerve is tightly compressed
nervous action does not go on, but electricity
can nevertheless pass/
G
82 SYLLOGISM. [CHAP.
Implies the following syllogism :
All tightly compressed nerves do not convey nervous
action ;
All tightly compressed nerves do convey electricity ;
Therefore, some things which convey electricity do not
convey nervous action.
The propositions are AAI in the third figure, i.e. the
syllogism is valid in Darapti. It is matter of further
inference that, because electricity is conducted by some
things which do not convey nervous action, therefore these
actions are not identical.
18. * With some of them God was not well
pleased ; for they were overthrown in the
wilderness. [w.]
An enthymeme of the first order, the major premise
being omitted. The order of statement is that by some
logicians called analytical, the conclusion being put first,
and the minor premise adduced as a reason or proof. The
major premise, assumed to be obvious, is to the effect that
* All who are overthrown in the wilderness are among those
with whom God is not well pleased. More fully stated,
indeed, the assumption might be that all who suffer from
any signal calamity are some of those with whom God is
not well pleased. To be overthrown in the wilderness is to
suffer from a signal calamity. This view makes a sorites
which the reader can put in order.
Ip. If the major term of a syllogism be the
predicate of the major premise, what do we
know about the minor premise ? [L.]
In answering syllogistic questions of this sort, great
VIIL] QUESTIONS AND ANSWERS. 83
attention must be given to throwing the reasoning into the
briefest and clearest form. Such questions, thus treated,
afford capital exercises in reasoning. The above question
may be answered thus :
If the major premise is affirmative its predicate, the major
term, is undistributed and must likewise be undistributed in
the conclusion; in this case the conclusion, and consequently
the minor premise, must be affirmative. If the major pre
mise be negative, then the minor premise must be affirmative,
in order to avoid negative premises ; thus in any case the
minor premise is affirmative. Or still more briefly thus :
The minor must be affirmative, for if negative then the
major would have to be affirmative, which would involve
Illicit Process of the Major.
20. Prove that O cannot be a premise in the
first or fourth figure ; and that it cannot be
the major in the second figure, or the minor
in the third. [M.]
If either of the premises be O, the conclusion must
be negative, so that its predicate the major term will be
distributed. But as O distributes only its predicate, and
the other premise, which must of course be affirmative and
universal, only distributes its subject, the syllogistic conditions
are much restricted. Thus OA as premises in the first
figure give an Undistributed Middle, the middle term being
subject of O and predicate of A. The premises AO give
Illicit Process of the Major Term, the predicate of A, the
major term, being undistributed. In the second figure O
cannot be the major, because its subject would then be the
major term, and undistributed. In the third figure it
cannot be the minor, because the major term would then
84 SYLLOGISM. [CHAP.
be predicate of A, the major premise, and thus Illicit Pro
cess of the Major would again arise. Finally, in the fourth
figure OA will give Illicit Major, and AO, Undistributed
Middle.
21. If it be known concerning a syllogism in the
Aristotelian system, that the middle term is
distributed in both premises, what can we infer
as to the conclusion ?
The syllogism cannot be in the second figure, because the
middle term, being the predicate in both premises, these
would both have to be negative, against Rule 5. In the
first figure the minor premise would have to be negative, in
order to distribute its predicate, the middle term ; but a
negative minor in the first figure gives Illicit Process of the
Major Term. In the third figure, however, the middle term
being subject of both premises will be twice distributed if
these be both universal, which happens in the moods Darapti
and Felapton. In the fourth figure the middle term is pre
dicate of the major and subject of the minor ; we must,
therefore, have a negative major and a universal affirmative
minor, which happens in the mood Fesapo. We find, then,
that a doubly distributed middle term can prove only a
particular conclusion, I or O, and these only in the third
and fourth figures.
22. Take an apparent syllogism subject to the
fallacy of negative premises, and inquire
whether you can correct the reasoning by
converting one or both of the premises into
the affirmative form.
[India Civil Service, July 1879.]
QUESTIONS AND ANSWERS. 85
Take premises in the first figure
No Fis Z;
No X is Y.
Obvert the major premise (see p. 42), and we have-
All Y is not-Z;
No X is F.
The premises would give no conclusion, the pseudo-mood
AEE in the first figure involving Illicit Process of the
Major. Obverting the minor, we have
No Fis Z;
All Xis not-F.
There are now four terms, and therefore no common
middle term at all. The reader may easily work out other
examples. (See Principles of Science, p. 62 ; first edition,
Vol. I. p. 75-)
23. Prove that the third figure must have an
affirmative minor premise, and a particular
conclusion.
In the third figure the major term is predicate of the
major premise. Now, if the minor premise be negative,
the conclusion will be negative (Rule 6), and distribute its
predicate the major term, but the major premise must be
affirmative in order to avoid negative premises. Thus, there
will arise Illicit Process of the Major Term. It follows, by
rednctio ad absurdum, that the minor premise cannot be
negative and must be affirmative.
Again the predicate of the minor premise is the minor
term, and, the premise being affirmative, this term will be
undistributed, giving a particular conclusion.
86 SYLLOGISM. [CHAP.
24. Show that if the conclusion of a syllogism be
a universal proposition, the middle term can
be but once distributed in the premises.
Questions of this sort can be most briefly answered by
counting the available number of distributed terms in the
premises. Thus, if the conclusion be a universal affirma
tive proposition, we need one distributed term for its
subject. But, as the premises must both be affirmative,
they contain at most two distributed terms, namely their
subjects. Hence there is only one place in which the
middle term can be distributed. On the other hand, if the
universal conclusion be negative, both major and minor
terms require to be distributed in the premises; but, as one
premise only can be negative, we cannot possibly have more
than three terms distributed, the subject and predicate of
the negative premise, and the subject of the affirmative one.
Two being required for the major and minor terms, there
remains only one distributed place for the middle term,
which was to be proved.
Observe that this result is the contrapositive of that
proved under Question 21 (p. 83).
25. Given the six rules of the syllogism, and
the rule that two particular premises prove
nothing, show that if one premise be par
ticular the conclusion must be particular.
This may be demonstrated by the following ingenious
reasoning of De Morgan (Formal Logic, p. 14).
If two propositions, P and Q, together prove a third, ,
it is plain that P and the denial of R prove the denial of
Q. For P and Q cannot be true together without ./?.
vin.] QUESTIONS AND ANSWERS. 87
Now, if possible, let P (a particular) and Q (a universal)
prove fi. (a universal). Then P (particular) and the denial
of R (particular) prove the denial of Q. But two parti
culars can prove nothing.
26. Show that the proposition O is seldom
admissible as a minor premise.
When O is the minor premise the conclusion must be
negative by Rule 6, and will therefore distribute its pre
dicate, the major term. As we must not have two negative
premises by Rule 5, the major premise must be affirmative,
and will not distribute its predicate. Hence the major
term must be the subject of the major premise. Now,
since the middle term becomes the undistributed predicate
of the major premise, it must be the predicate of the minor
premise, in order that it may be once distributed. Thus
we conclude that O can be the minor premise only in the
second figure, giving the mood Baroko.
27. Show that a universal negative proposition
(E) is highly efficient as a major premise.
M
Since E has both its terms distributed, either of them
may serve as the major term, which, the conclusion being
negative, must be distributed. The other term will then
serve to distribute the middle term once. The minor pre
mise may therefore be chosen at will, provided that it be
affirmative in order to avoid negative premises. There is
no restriction of figure, and accordingly we find valid moods
with E as major premise in all of the four figures ; in fact,
no less than eight of the nineteen recognised moods begin
with E.
88 SYLLOGISM. [CHAP.
28. Name the weakened moods of the syllogism.
In what figure can there be no weakened
mood, and why ? Do any of the nineteen
moods commonly recognised give a weaker
conclusion than the premises would warrant ?
By a weakened mood is meant one which gives a particular
conclusion when a universal conclusion might have been
drawn. The information obtained from the premises is
thus weakened. This can, of course, happen only when
the conclusion of the stronger mood is universal. Hence,
in the third figure, which gives only particular conclusions,
there can be no weakened mood. In the other figures each
mood which has a universal conclusion will have a corre
sponding weakened mood with conclusion of the same
quality. Thus Barbara gives a mood A A I ; Celarent,
EAO; Cesare, EAO; Camestres, AEO; in the fourth
figure only Camenes admits of a weakened form, AEO.
Thus the weakened moods are five in number.
Bramantip of the fourth figure is the single mood alluded
to in the latter part of the question.
Considering that it is impossible to employ conversion
by limitation without weakening the logical force of the
premise, it is too bad of the Aristotelian logicians to slight
the weakened moods of the syllogism as they have usually
done.
29. Can we under any circumstances infer a
relation between X and Z from the pre
mises
Some Fs are Xs ;
Some Fs are Zs ?
[India Civil Service, July 1879.]
viii.] QUESTIONS AND ANSWERS. 89
Not if some Fs bear the sense attributed to the ex
pression in Logic. The indefinite adjective of quantity
some is so indefinite, that it must never be interpreted twice
over with the same meaning. But if the some Y in the
one premise were intended by the arguer to be the same
some V as in the other premise, the term would practically
become a distributed one, and the premises might give a
valid conclusion in the mood Darapti. Dr. Thomson has
remarked (Laws of Thought, 77, p. 132), that the word
(some) appears to be employed in the two senses of
" Some or other," and "Some certain," in common language.
Observe, however, that it is in the former purely indefinite
sense that logicians have always used the word, so that
some K* must not be identified with some K
30. Is the following argument a valid syllogism ?
That which has no parts cannot perish by the
dissolution of its parts ; the soul has no
parts ; therefore, the soul cannot perish by
the dissolution of its parts.
This example is quoted from the Port Royal Logic, Part
III. Chap, ix., Example 6. It is there remarked that
several persons advance such syllogisms in order to show
the inaccuracy of the unconditional rule (5) that nothing
can be inferred from negative premises. Without re
membering what was said in the Art of Thinking, I made
the same objection in the Principles of Science, p. 63 (first
ed. Vol. I. p. 76), and I must still hold that in its bare
statement the syllogistic rule is actually falsified. But it
must, no doubt, be allowed that if the premises are to
be treated as both negative, then there are four terms ; the
middle term is broken up into two terms, that which has
90 SYLLOGISM. [CHAP.
not parts/ and that which has parts ; the soul is denied
to be the latter ; the former is that of which it is asserted
negatively that it cannot perish, etc. It comes simply to
this, that the syllogistic rules are to be interpreted as a
whole, and in making the above example conform to the
first rule (see p. 71) we make it conform also to the fifth
rule.
Professor Groom Robertson has criticised my treatment
of this subject (Mind, 1876, p. 218, note}, urging that
1 There are four terms in the example, and thus no syllogism,
if the premises are taken as negative propositions ; while
the minor premise is an affirmative proposition, if the terms
are made of the requisite number three. No doubt
Professor Robertson is substantially right, but it may be
noticed that my words were so cautious as hardly to commit
me to an erroneous statement. I now find that the point
has been treated by many logicians in addition to those of
Port Royal, as for instance, Burgersdicius ; De Morgan,
Formal Logic, p. 139, Art. 3; Bain, Deductive Logic, p. 164;
Devey, Logic, or the Science of Inference, 1854, p. 129;
Essai sur la Logique, 1763, p. 106.
31. In reference to the syllogism, Mr. Jevons
urges that it sometimes yields a conclusion
that is open to misinterpretation, as in the
example-
Potassium is a metal ;
Potassium floats on water ;
Therefore, Some metal floats on water.
Examine this criticism carefully.
[Moral Science Tripos, Cambridge, Dec. 1876.]
I said in the Principles of Science (pp. 59-60; first ed.
QUESTIONS AND ANSWERS. 91
Vol. I. pp. 71-2), that my inference, namely, Potassium
metal = potassium floating on water, is of a more exact
character than the Aristotelian result Some metal floats on
water. The some after all is only here an indefinite
name for potassium, and unless we constantly bear in
mind that some means in logic one, and it may be more
or all, the reasoner is apt to confuse some with the
plural several. This view of the matter was criticised by
Professor Groom Robertson (Mind, 1876, p. 219).
32. What is the nature of the argument, if any,
in the apparent enthymeme, The field is
neglected because the soil is poor ?
This may, of course, be an argument in Barbara, thus
Every field of poor soil is neglected ;
This is a field of poor soil ;
. . This field is neglected.
But the statement may also mean that the soil being poor
is the reason or cause why the owner neglects it ; in this
case, it is not an argument but a causal relation. The
student, therefore, must always look out for ambiguities in
the conjunction for, because, etc., which may certainly
bear one of two if not of more senses. The relation between
premises and conclusion has nothing whatever to do with
the relation between cause and effect.
33. Explain It is scarcely ever possible de
cidedly to affirm that an} argument involves
a bad syllogism ; but this detracts nothing
from the value of the syllogistic rules. [R.]
Scarcely any one in ordinary writing or discourse states a
syllogism in full form ; it is always presumed that the hearer
92 SYLLOGISM. [CHAP.
or reader is enough of a logician to supply what is wanting.
Now the missing premise may generally be supplied in such
a way as to make a good syllogism formally speaking, that
is to say, so as to avoid any breach of the syllogistic rules.
It is another matter whether the new premise is materially
true. The value of the syllogistic rules is, then, that they
enable us to assign the premises which would be requisite
to support the conclusion put forward. They thus oblige
the arguer to define the nature of his assumptions, or else
to yield up his conclusion.
34. How shall we reduce the following syllogism
to the first figure ?
All men are liable to err ;
None who are liable to err should refuse advice :
None who should refuse advice are men.
This argument is in the absurd fourth figure, in the mood
Camenes. In this name the letter m directs us to transpose
the premises, and the final s directs us to convert the con
clusion simply ; making these changes, we obtain the same
argument in the more natural form of Celarent, thus
None who are liable to err should refuse advice ;
. -.*.
All men are liable to err ;
No men should refuse advice. ^
35. How shall we reduce the following syllogism
to the first figure ?
All birds are vertebrates ;
Some winged animals are not vertebrates ;
Some winged animals are not birds.
The premises are A O in the second figure, and the con
clusion being O, the argument is a valid syllogism in Baroko.
VIIT.] QUESTIONS AND ANSWERS. 93
The letter k directs us to employ the Reductio ad
impossibile, as explained in the Elementary Lessons, p. 149.
Or we may convert the major by contraposition, getting
All not-vertebrates are not birds ;
Some winged animals are not-vertebrates ;
Therefore, Some winged animals are not birds.
Taking the negative term not-vertebrates as the middle
term, this is valid in Barbara.
36. Can we reduce the mood Camestres per
impossibile ?
Taking the symbolic example
All Xs are Ys ;
No Zs are Fs ;
Therefore, No Zs are Xs,
and assuming for sake of argument that the conclusion is
false, the contradictory * some Zs are Xs will be true, which
put as minor premise with the original as major, gives the
valid syllogism in Darii
All ^s are Fs ;
Some Zs are Xs ;
Therefore, Some Zs are Fs.
But this conclusion is the contradictory of the original
minor premise no Zs are Fs, so that we cannot contradict
the conclusion of Camestres without producing a syllogism
in Darii to contradict one of our original premises. Thus
we prove the conclusion of Camestres indirectly by a mood
of the first figure. It will be found on trial that all the
moods of the imperfect figures may be similarly proved
indirectly by one or other of the moods of the first or
so-called perfect figure,
CHAPTER IX.
QUESTIONS AND EXERCISES ON THE SYLLOGISM.
i. ASSIGN the moods of the following valid syllogisms,
pointing out in succession
(a) The conclusion ;
(b) The middle term ;
(c) The major term, and the major premise containing it ;
(d) The minor term, and the minor premise containing it ;
(e) The quantities and qualities of the three propositions ;
(/) Their symbols ;
(g) The order in which they should be technically placed ;
(h) The figure of the syllogism ;
(/) The mood, and its mnemonic name.
(1) No birds are viviparous ;
All feathered animals are birds ;
No feathered animals are viviparous.
(2) Robinson is plain spoken ; for he is a Yorkshire man,
and all Yorkshire men are plain spoken.
(3) Birds are not viviparous animals ;
Bats are viviparous animals ;
Bats, therefore, are not birds.
(4) Whatever investigates natural laws is a science ;
Logic investigates natural laws ;
| Logic is a science.
CHAP, ix.] QUESTIONS AND EXERCISES. 95
(5) Quicksilver is liquid at ordinary temperatures ;
Quicksilver is a metal ;
Some metal, therefore, is liquid at ordinary tempera
tures.
(6) True fishes respire water containing air ;
Whales do not respire water containing air ;
Whales, therefore, are not true fishes.
2. Arrange the following valid syllogisms in the usual
strict order of major premise, minor premise and conclusion.
Name the figure and mood to which they belong. In exa
mining syllogisms, always follow the directions of the first
question.
(1) Iridium must be lustrous; for it is a metal, and all
metals are lustrous.
(2) Some pleasures are not praiseworthy; hence some
pleasures are not virtuous, for whatever is not
praiseworthy is not virtuous.
(3) Epicureans do not hold that virtue is the chief good,
but all true philosophers do hold that it is so;
accordingly, epicureans are not true philosophers.
(4) Some towns in Lancashire are unhealthy, because
they are badly drained, and such towns are all
unhealthy.
3. Draw conclusions from the following pairs of premises,
specifying the figure and mood employed
,,( Every virtue is accompanied with discretion;
1 There is a zeal without discretion.
. ^ f Sodium is a metal ;
1 Sodium is not a very dense substance.
All lions are carnivorous animals ;
No carnivorous animals are devoid of claws,
96 SYLLOGISM. [CHAP.
( Combustion is chemical union ;
(4) -] Combustion is always accompanied by evolution
(. of heat.
C All boys in the third form learn algebra ;
(5) \ There are no boys in the third form under twelve
V years of age.
Nihil erat quod non tetigit :
Nihil quod tetigit non ornavitj
4. Examine the following arguments and point out which
are valid syllogisms, naming the figure and mood as before ;
in the case of such as are pseudo-syllogisms, name the rule
of the syllogism which is broken thereby, and give the
technical name of the fallacy
(1) All feathered animals are vertebrates ;
No reptiles are feathered animals ;
Some reptiles are not vertebrates.
(2) Some vertebrates are bipeds ;
Some bipeds are birds ;
Some birds are vertebrates.
(3) All vices are reprehensible ;
Emulation is not reprehensible ;
Emulation is not a vice.
(4) All vices are reprehensible ;
Emulation is not a vice ;
Emulation is not reprehensible. [L.]
(5) Some works of art are useful ;
All works of man are works of art ;
Therefore some works of man are useful [L.]
(6) Iron is a metal ;
All metals are soluble ;
Iron is soluble.
ix.] QUESTIONS AND EXERCISES. 97
(7) Aryans are destined to possess the world ;
Chinese are not Aryans ;
Chinese are not destined to possess the woWd.
(8) Only ten-pound householders have votes ;
Smith is a ten-pound householder ;
Smith has a vote.
5. What are the suppressed premises which are evidently
presumed to exist by those who set forth the following
imperfectly stated syllogisms? State figure and mood as
usual.
(1) Blessed are the meek : for they shall inherit the earth.
(2) This iron is not malleable ; for it is cast iron.
(3) Whosoever loveth wine shall not be trusted of any
man ; for he cannot keep a secret.
(4) Being born in Africa, he was naturally black.
(5) Some parallelograms are not regular plane figures, for
they cannot be inscribed in a circle.
(6) Suffer little children to come unto me ; for of them is
the kingdom of Heaven.
(7) It is dangerous to tell people that the laws are not
just ; for they only obey laws because they think
them just.
(8) The line A B is equal to the line CD; for they are
both radii of the same circle.
(9) Whales are not true fishes, for they respire air; more
over they suckle their young.
(10) The Queen is at Windsor, for the royal standard is
flying,
(n) The science of logic is very useful; it enables us to
detect our adversaries fallacies.
(12) He must be in York, for he is not in London.
98 SYLLOGISM. [CHAT.
(13) I shall not derive my opinions from books for I
have none. [Mansfield, H. of L., 1780.]
(14) The nation has a right to good government; there
fore it may rebel against bad governors.
(15) The wise man has an infinity of pleasures ; for virtue
has its delights in the midst of the severities that
attend it.
6. Point out which of the following pairs of premises will
give a syllogistic conclusion, and name the obstacle which
exists in other cases.
(1) No A is B ; some B is not C.
(2) No A is B \ some not C is B.
(3) All B is not A ; some not A is B.
(4) Some not A is B ; no C is B.
(5) All not B is C; some not A is B.
(6) All A is B \ all not C is not B.
(7) All not B is not C all not A is not B.
(8) All ^ is not ^ ; no B is not C.
(9) All C is not B ; no ^ is not ./?.
7. To what moods do the following belong?
(1) < All ^ is A ; only C is A ; therefore only C is B:
(2) l All B is ^4 ; nothing but C is ^ ; therefore nothing
but Cvs. B:
See Dante s De Monarchia, as translated by F. C. Church,
and appended to the Essay on Dante, by the Rev. R. W.
Church, 1878, p. 195. Many curious specimens of reason
ing, sometimes pedantic, might be drawn from the De
Monarchia.
8. Supply premises to prove or disprove the following
conclusions
ix.] QUESTIONS AND EXERCISES. 99
(1) The loss of the Captain proves that turret-ships are
not sea-worthy.
(2) The cottage-hospital system should be adopted.
(3) The Prussians are justified in refusing the rights of
war to Garibaldi if they find him fighting against
them. [E.]
(4) Private property should be respected in war.
(5) No woman ought to be admitted to the franchise, [o.]
(6) The law of libel requires to be amended. [o.]
(7) Capital punishment ought to be abolished. [o.]
(8) Royal parks ought not to be used for political meet
ings, [o.]
(9) Written examinations are not a safe test of merit.
(10) Written examinations are a safe test of merit. [E.]
(n) The Annuity-tax should be done away with. [E.]
(12) Any national system of education should be a secular
system. [E.]
9. In how many different moods may the argument im
plied in the following question be stated ? No one can
maintain that all persecution is justifiable who admits that
persecution is sometimes ineffective.
How would the formal correctness of the reasoning be
affected by reading deny for * maintain ? [c.]
i o. What conclusions, and of what mood and figure, can
be drawn from each pair of the following propositions ?
(1) None but gentlemen are members of the club.
(2) Some members of the club are not officers.
(3) All officers are invited to dine.
(4) All members of the club are invited to dine, [c.]
ii. Express the following reasonings in each of the four
syllogistic figures.
ioo SYLLOGISM. [CHAP.
(1) Some medicines should not be sold without registering
the buyer s name, for they are poisons. [E.]
(2) No unwise man can be trusted ; hence some specula
tive men are unworthy of trust, for they are
unwise. [E.]
12. Can the following argument be stated in the form of
a syllogism, and if so, what is the middle term ?
The power of ridicule is a dangerous faculty, since it
tempts its possessor to find fault unjustly, and to distress
some for the gratification of others.
13. If the proposition warmth is essential to growth
occurred as the premise of a syllogism, would you treat
warmth as a distributed or an undistributed term ? [E.]
14. Show that the following single propositions may be
regarded as enthymemes, that is as equivalent to imperfectly
expressed syllogisms :
(1) Have thou nothing to do with that just man. [w.]
(2) If wishes were horses, beggars would ride.
(3) Large colonies are as detrimental to the power of a
State, as overgrown limbs to the vigour of the
human body.
(4) If I had read as much as my neighbours, I would
have been as ignorant. [HOBBES.]
(5) All law is an abridgment of liberty and consequently
of happiness.
(6) Thales being asked what was the most universally
enjoyed of all things, answered Hope ; for they
have it who have nothing else.
(7) I will give thee my daughter if thou canst touch
heaven.
ix.] QUESTIONS AND EXERCISES. 101
(8) If all the absurd theories of lawyers and divines were
to vitiate the objects in which they are conversant,
we should have no law and no religion left in the
world. [BURKE.]
15. Distinguish between the causal, simply logical, or
other, senses of the copulative conjunctions in the
following
(1) It will certainly rain, for the sky looks black.
(2) The people are happy because the government is
good.
(3) This plant is not a rose ; for it is monopetalous.
(4) The ancient Romans trusted their soothsayers, and
must therefore have been frequently deceived.
(5) A favourable state of the exchanges will lead to im
portation of gold : this will cause a corresponding
issue of bank-notes which will occasion an ad
vance in prices ; which again will check exportation
and encourage importation, tending to turn the
exchanges against us. [GILBART, 1851, p. 284.]
1 6. Form an example of a syllogism in which there are
two prosyllogisms, one attached to the middle and the
other to the minor term. [H.]
1 7. Prove that a valid sorites with n premises must have
;/ + i terms, and is capable of giving n ^ n ~ * conclusions.
2
1 8. Can the following Shakspearean passage (Hamlet,
Act v. Scene i.) be stated in the form of a sorites ?
Alexander died, Alexander was buried, Alexander
returneth into dust ; the dust is earth ; of earth we make
loam ; and why of that loam, whereto he was converted,
might they not stop a beer barrel?
102 SYLLOGISM. [CHAP. ix.
19. Throw the reasoning of the following passage into
syllogistic form :
Carbon, which is one of the main sources of the
nourishment of plants, cannot be dissolved in water in its
simple form, and cannot therefore be absorbed in that form
by plants, since the cells absorb only dissolved substances.
All the carbon found in plants must consequently have
entered them in a form soluble in water, and this we find in
carbonic acid, which consists of carbon and oxygen. [A.]
20. Complete such of the following arguments as may be
considered sound but incomplete syllogisms :
(1) The people of the country are suffering from famine,
and as you are one of the people of the country,
you must be suffering from famine.
(2) Light cannot consist of material particles, for it does
not possess momentum.
(3) Aristotle must have been a man of extraordinary
industry; for he could not otherwise have pro
duced so many works.
(4) Marcus Aurelius was both a good man and an
Emperor ; hence it follows that Emperors may be
good men, and vice versa.
(5) Nothing which is unattainable without labour is
valuable; some knowledge is not attainable with
labour, and is therefore valuable.
(6) All gasteropods are mollusks, and no vertebrate
animals are mollusks ; therefore no gasteropods
are vertebrate.
(7) Suicide is not always to be condemned ; for it is but
voluntary death, and voluntary death has been
gladly embraced by many great heroes.
CHAPTER X.
TECHNICAL EXERCISES IN THE SYLLOGISM.
1. PROVE, from the general rules of Syllogism, that when
the major term is predicate in its premise, the minor
premise must be affirmative.
2. Prove that,, when the minor term is predicate in its
premise, the conclusion cannot be a universal affirmative.
>]
3. Prove that there must always be in the premises one
distributed term more than in the conclusion.
4. Prove that the major premise of a syllogism, whose
conclusion is negative, can never be a particular affirmative.
5. Prove that when the minor premise is universal nega
tive, the conclusion (unless weakened) will be universal.
6. Prove that, if in the first figure we transpose the major
premise and conclusion, we obtain a pseudo-mood.
7. In the third figure, if the conclusion be substituted for
the major premise, what will the figure be ? [BAGOT.]
8. Prove that no syllogism in the fourth figure can be
correct which has a particular negative among its premises,
or a universal affirmative for its conclusion. [L.]
9. If the major term be universal in the premises and
particular in the conclusion, determine the mood and figure,
it being understood that the conclusion is not a weakened
one. [c.]
io 4 SYLLOGISM. [CHAP.
10. Why is it impossible to transform the mood A E O
from the second figure into the first ?
1 1. What figure must have a negative conclusion ? Why
must it?
12. What figure must have a particular conclusion?
Why must it ?
13. Why must the major premise of the fourth figure not
be O?
14. Why must the minor premise of the fourth figure
not be O ?
15. If the minor premise of the first figure were not
affirmative, what fallacy would be committed ?
1 6. If the major premise of the first figure were I, what
fallacy would be committed ?
17. What kind of proposition does not occur in the
premises of the first figure ? Why does it not occur ?
1 8. If the major premise of the second figure were
particular, what fallacy would be committed ?
19. What is remarkable about the conclusions of the
third figure ?
20. What kind of proposition cannot be proved by the
fourth figure?
21. In what mood of the syllogism can a subaltern pro
position be substituted for its subalternans (universal of
same quality) as premise without affecting the conclusion ?
22. If one premise of a syllogism be O, what must the
conclusion be ?
23. Prove that a universal affirmative proposition can
form the conclusion of the first figure alone.
24. Why is O A O excluded from the first and second
figures ?
25. Why is it that the moods E A O and E I O are true
in all the four figures ?
x.] TECHNICAL EXERCISES. 105
26. It is said that A is the most difficult conclusion to
establish by the syllogism, and the most easy to overthrow ;
O, on the contrary, is the most easy to establish, and the
most difficult to overthrow. What is there in the moods of
the syllogism to support this view ?
27. When both premises of an apparent syllogism are
negative, the real middle term is an external sphere, and is
consequently undistributed [SOLLY]. Explain the meaning
of this statement.
2 8. If the minor premise of a syllogism be O, what is the
figure and mood?
2 9. Prove that there cannot be more than four figures of
the syllogism.
30. If one premise be O, what must the other be?
31. Show that in the fourth figure the conclusion may
be either affirmative or negative, and, if negative, either
universal or particular.
32. Given the major premise particular, or the minor
premise negative, what must the other premise be ? Why
so? [P.]
33. If the minor term be predicate of minor, or major
subject of major, the conclusion may not be A? [p.]
34. Prove that the combination of a particular major
premise with a negative minor premise leads to no valid
inference.
35. Prove that wherever there is a particular conclu
sion without a particular premise, something superfluous is
invariably assumed in the premises.
36. Determine in what affirmative moods the middle term
may be universal in the major premise, and particular in the
minor.
37. Determine in what negative moods the same may
occur. [pi
io6 SYLLOGISM. [CHAP. x.
38. Determine how many universal terms may be in the
premises more than in the conclusion.
39. Determine how many particular terms may be in the
premises more than in the conclusion.
40. Determine in what cases there may be in a syllogism
an equal number of universal terms and of particular, [p.]
41. How do you reduce Camestres, Festino, Darapti,
Fresison to the first figure ?
42. Exemplify the reduction of Baroko and Bokardo by
the process per impossibile.
43. Show that Cesare, Disamis, Camenes, and other
moods can likewise be proved per impossibile. (See Kars-
lake, 1851, Vol. I. p. 81.)
44. Reduce Celarent to the fourth figure. To how
many other figures can you reduce it ?
45. Reduce Felapton to the second figure.
46. To what other moods, respectively, can you reduce
Darii, Ferio, Barbara?
CHAPTER XL
CUNYNGHAME S SYLLOGISTIC CARDS.
1. IN this age of mechanical progress it may be a matter
of surprise that no one has produced a syllogistic machine.
About two centuries ago Pascal and Leibnitz invented true
calculating machines, and Swift, incited perhaps by the
accounts of these machines, described the professors of
Laputa as in the possession of a thinking machine. About
thirty years ago the late Alfred Smee, F.R.S., proposed the
construction of a kind of mechanical dictionary, together
with a contrivance for comparing the ideas defined in it.
More recently I have constructed a machine which analyses
the meaning of any propositions worked upon its keys,
provided they do not involve more than four distinct terms.
Yet the rules of the syllogism have never been put into a
mechanical form. So much the worse for the rules of the
syllogism.
2. Some approximation to a syllogistic machine has,
however, been recently made by Mr. Henry Cunynghame,
B.A. He has devised certain cards, which, if placed one
upon the other, infallibly give a syllogistic mood when
that is possible, and when it is not, indicate the absence
of a conclusion. The contents of the cards, too, can be
condensed into a hollow cylinder turning upon a solid
cylinder, in such a way as to give all possible syllogistic
io8 SYLLOGISTIC CARDS. [CHAP.
moods in one turn of the handle. This device, though
hardly perhaps to be called a syllogistic machine, is probably
the nearest approximation to such a machine which is pos
sible. I am now enabled, by Mr. Cunynghame s kindness,
to describe these ingenious and interesting devices for the
first time.
3. The syllogistic cards consist of a set of eight larger
and a set of eight smaller cards. Each larger card is 3^
inches high by 2\ inches broad, and bears, near to its upper
edge, one of the eight possible propositions connecting M,
the middle term, with />, the major term. There are, of
course, four propositions of the forms A, E, I, O, in which
M is the subject, and four more in which P is the subject.
In the lower part of each larger card is written, in a certain
position, each proposition between S and P, which can
possibly be drawn from a syllogism having the proposition
on the upper edge of the card for its major premise. Thus,
under the major premise * Some M is P] we find Some S
is P, as the only conclusion that can be drawn from it ;
but the universal affirmative proposition, All P is J/,
admits of any one of three possible conclusions, namely,
No S is P; Some S is P, or some S is not P. 1
4. Each card of the lesser set is also 2-J- inches wide, but
only 3 inches high, and bears on its upper edge one of the
eight possible minor premises connecting S, the minor term,
with M, the middle term. In the lower part of each smaller,
or, as we may call it, minor card (with one exception), is cut
one, or in some cases two rectangular openings, so adjusted
that if any minor card be placed upon a larger or major
card, so that the major and minor premises inscribed upon
them be visible, one below the other, the conclusion, if any,
belonging to those premises is seen through the opening in
the minor card. If two conclusions, a universal proposition,
XL] SYLLOGISTIC CARDS. 109
and its subaltern minor, are both possible, both will be seen.
Thus, if we take the major Some P is MJ and the minor
All M is SJ we see below the conclusion * Some S is P, 1
the mood being Dimaris in the fourth figure. If we take
the minor All S is J/, and place it upon the major No
M is P, we see below No S is P, the correct conclusion
in the mood Celarent, together with Some S is not P, the
corresponding weakened or subaltern conclusion. If the
major be Some P is not M, or the minor Some M is
not 5, no conclusion can appear at all.
5. The cards are shown in complete detail on the next
page. The principle on which they are constructed is that
of excluding illogical conclusions. No conclusion is written
on the major card but such as the premise at the top will
warrant, and no conclusion is left uncovered by the minor
card if it be unwarranted by the minor premise. The
student can readily cut out cards according to the directions
given above and the figures on the next page, but they
must be cut exactly to scale. It will assist the con
struction if each major card be divided by a pencil mark
into seven horizontal spaces, each half an inch high, each
minor card being divided into six similar spaces.
6. The syllogistic cylinder is more compact and handy,
but much less easy to describe and illustrate. The prin
ciple is exactly the same, but considerable ingenuity was
needed to combine the cards together into cylinders in the
best way. The order of the minor premises on the in
ferior or moving cylinder is as follows : All S is MJ
1 All M is S, Some S is M; Some M is S t > No S is M,
No M is SJ Some S is not MJ Some M is not
The major premises are in like order, All P is M, All
; Some -Pis M; etc.
CUNYNGHAME S SYLLOGISTIC CARDS.
MAJOR CARDS.
All M is P
All S is P
Some S is P
No M is P
NoSisP
Some 6" is not F
Some M is P
Some S is P
Some ^f is not P
Some 6" is not P
All P is M
No 6 1 is P
Some S is P
Some vS" is not P
No /> is M
NoSisP
Some vS" is not P
Some PisM
Some S is />
Some /> is not M
MINOR CARDS.
All S is ^/
No 6" is M
Some 6" is M
So me 6" is not M
All J/ is 6 1
No Af is 6
Some M is 6"
Some Af is not ,5"
L
CHAPTER XII.
FORMAL AND MATERIAL TRUTH AND FALSITY.
i. THE rules of syllogistic inference teach us how,
from certain premises assumed to be true, to draw other
propositions which will be true under those assumptions.
But if, instead of supposing the premises to be true, we
regard them as materially false, various puzzling questions
arise as to the conclusions which may properly be drawn.
Such questions have not been adequately treated in any
popular manual of logic and as they lead to results of
great practical importance, and at the same time furnish
admirable exercises in the discrimination of good and bad
reasoning, I propose to draw special attention to this
subject in the following questions with answers.
2. Is it possible to draw a false conclusion from
true premises ?
It is possible, of course, to draw any conclusion from any
premises, if we disregard the principles of logic and of
common sense. But when we speak in a logical work of
drawing a conclusion, we must be understood to mean
drawing a conclusion logically, in accordance with the Laws
of Thought and the Rules of the Syllogism. Now the nature
of the logical relation between premises and conclusion is
ii2 TRUTH AND FALSITY. [CHAP.
this, that if the premises are true the conclusion is true ;
truth is, as it were, carried from the premises into the
conclusion ; not the whole truth necessarily, but nothing
except truth. The question above must, of course, be
answered with a direct negative.
3. Is it possible to prove a true conclusion with
false premises ?
To prove a true conclusion, or to prove that a certain
conclusion is true, must mean to establish its truth in
the opinion of the persons concerned. To prove,
says Wesley, 1832, p. 90, is to adduce premises which
establish the truth of some conclusion. Now, the rela
tion of premises and conclusion in a syllogism, as stated
just above, is that if the premises are true the conclusion
must be admitted to be likewise true. But, if certain
persons regard the premises as false, they cannot possibly
regard such premises as establishing or proving the truth of
the conclusion. Solly, indeed, points out (1839, note, p. 9)
the possibility of a true conclusion from false premises in
every form of reasoning. But this remark can only mean
that the conclusion is materially true, or known to be true,
on other grounds.
4. If the premises of a syllogism are false, does
this make the reasoning false ?
No. The reasoning is correct if the form of the pre
mises and conclusion agree with that of any valid mood of
the syllogism or other development of the Laws of Thought,
wholly regardless of the material truth of any of the propo
sitions per se. The most ridiculous proposition may make
a good syllogism : for instance
xii.] QUESTIONS AND ANSWERS. 113
Every griffin has angles equal to two right angles;
Every triangle is a griffin ;
Therefore, Every triangle has, etc.
This is, of course, a valid syllogism in Barbara, and if the
premises were true the conclusion would be true ; the pre
mises being untrue, the truth of the conclusion is entirely
unaffected by the reasoning.
5. (a) The most perfect logic will not serve a
man who starts from a false premise.
(//) I am enough of a logician to know that
from false premises it is impossible to draw a
true conclusion.
Comment carefully upon the two foregoing
extracts.
Both the above sentences have been written in perfect
seriousness by men of intelligence, and they are fair speci
mens of the logic which would pass muster almost anywhere
except in a book of logical exercises. In the first place, as
regards (a), if a premise be materially false it cannot give
a conclusion materially true ; accidentally, indeed, it might
do so by paralogism ; but, as the logic is assumed to be
most perfect, we are dealing only with material truth and
falsity. Nevertheless, the logic may serve the man well ;
for he can learn the truth or falsity of his propositions
by observing their congruity with external facts. Now, if
he has failed to learn in this way directly the falsity of his
premise, his only chance is to draw logical conclusions from
that premise, and then observe whether they are or are not
materially verified. It is quite clear that if by correct logic
we reach a conclusion materially false, then we must have
started from premises which involved material error. This
1 1 4 TRUTH AND FALSITY. [CHAP.
procedure represents, in fact, the real method of induction,
as the inverse process of deduction , by which we learn all the
more complicated truths of physical and moral science. (See
Principles of Science, Chaps. XI. and XII). The sentence
(a\ then, is true only on the supposition that a man, having
adopted a false premise, will blindly accept all its false
results, that is to say, will reason in a purely deductive
manner.
The sentence (b) is erroneous, because, as we have fully
learnt (p. 112), we can from false premises draw a true
conclusion in good logical form. But there was doubtless
confusion in the writer s mind between formal and material
falseness, and had he said that by premise^ materially false
it is impossible to establish the material truth of any con
clusion, he would have been correct.
6. An apparent syllogism of the second figure
being examined is found to break the rules of
the syllogism, the middle term being undis
tributed. On further examination it is re
marked that one of the premises is evidently
false, and the other true. What can we infer
from such circumstances concerning the truth
or falsity of the conclusion ? [c.]
As in the second figure the middle term is predicate in
both the premises, the apparent syllogism can break the
third rule of the syllogism, requiring that it shall be distri
buted once at least, only when both the premises are affirma
tive. The premises must therefore be A A, A I, I A, or II.
In the first case, if the premises be
All ^s are Fs,
All Zs are Fs,
xii.] QUESTIONS AND ANSWERS. 115
and we assume the first one to be false, we obtain its contra
dictory as true ; thus
Some Xs are not Fs ;
All Zs are Fs.
The conclusion must, by Rule 6, be negative, and there
will be Illicit Process of the Major. Assuming the second
premise to be false, we get
All Xs are Fs ;
Some Zs are not Fs.
Whence we may correctly infer in the mood Baroko,
Some Zs are not Xs.
In the case of A I, we obviously cannot assume A to be
false ; but if I be false, we get
All Xs are Fs ;
No Zs are Fs ;
Therefore, No Zs are Xs.
In the premises I A, we cannot assume A to be false without
Illicit Process of the Major Term ; but if I be false, we
have Cesare. Lastly, in 1 1 only the major can be taken
as false, and its contradictory then gives us a syllogism in
Festino. If the conclusion of the apparent or pseudo-
syllogism in question does not correspond with what we
thus obtain, the conclusion is logically false as compared
with the new premises assumed.
7. If (i) it is false that whenever X is found F
is found with it, and (2) not less untrue that
X is sometimes found without the accompani
ment of Z, are you justified in denying that
(3) whenever Z is found there also you may
n6 TRUTH AND FALSITY. [CHAP.
be sure of finding Y? And however this may
be, can you in the same circumstances judge
anything about Y in terms of Z ? [R.]
This excellent example of reasoning by contradictories
can be easily solved by adhering to the simple rules of
opposition, and gradually undoing the perplexities. The
supposition that, whenever X is found Y is found with it,
may be stated as the universal affirmative all Xs are Ys, ;
but as this is false, its contradictory some Xs are not Ys
is the true condition. That (2) X is sometimes found with
out the accompaniment of Z, would mean that some Xs
are not Zs ; but, being asserted to be untrue, the real
condition is its contradictory All Xs are Zs. Thirdly,
whenever Z is found, there also you may be sure 01 rinding
y, means that all Zs are ys ; but, if you deny this, you
must assert that some Zs are not ys. Putting these
propositions together, thus
1 i ) Some Xs are not ys ;
(2) A\\Xs areZs;
Hence, (3) Some Zs are not ys ;
we find that they make a valid syllogism in the third
figure, and the mood Bokardo. The conclusion, being a
particular negative, cannot be converted directly ; we can
only obtain by obversion and conversion some not-ys
are Xs. Thus we must, I presume, answer the last part
of the problem negatively.
8. What is the precise meaning of the assertion
that a proposition say All grasses are edible
is false ?
The doctrine of the falsity of propositions is generally
xii.] QUESTIONS AND ANSWERS. 117
supposed to be defined with precision in the ancient formula
of the square of opposition. If a universal affirmative pro
position is false, its contradictory, the particular negative, is
true, so that, in the case of the example given above, we
infer, that some grasses are not edible. Similarly, from
the falsity of E we infer the truth of I, and vice versa.
But it does not seem to have occurred to logicians in general
to inquire how far similar relations could be detected in the
case of disjunctive and other more complicated kinds of
propositions. Take, for instance, the assertion that All
endogens are all parallel-leaved plants. If this be false,
what is true ? Apparently that one or more endogens are
not parallel-leaved plants, as, else that one or more parallel-
leaved plants are not endogens. But it may also happen
that no endogen is a parallel-leaved plant at all. There are
three alternatives, and the simple falsity of the original does
not show which of the possible contradictories is true.
But the question arises whether there is not confusion
of ideas in the usual treatment of this ancient doctrine of
opposition, and whether a contradictory of a proposition is
not any proposition which involves the falsity of the original,
but is not the sole condition of it. I apprehend that any
assertion is false which is made without sufficient grounds.
It is false to assert that the hidden side of the moon is
covered with mountains, not because we can prove the con
tradictory, but because we know that the assertor must have
made the assertion without evidence. If a person ignorant
of mathematics were to assert that all involutes are tran
scendental curves, he would be making a false assertion,
because, whether they are so or not, he cannot know it.
Professor F. W. Newman has correctly remarked that no
one can really believe a proposition the terms of which he
does not understand (Lectures on Logic, 1838, pp. 35, 36).
n8 TRUTH AND FALSITY. [CHAP.
This is unquestionably true; for, if he does not know
what things he is speaking about, he cannot possibly bring
them to comparison in his mind. A witness who swears that
a prisoner did a certain act when, as a matter of fact, he
does not know whether the prisoner did it or not, swears
falsely, independently of the question whether rebutting
evidence can be brought to prove the perjury. It is reported
that a man, who wished to be thought an acquaintance of
Dr. Johnson, remarked to him in coming out of church, A
good sermon to-day, Dr. Johnson. That may be, sir,
replied the very much over-estimated doctor, but I m not
sure that you can know it. This hits the point precisely.
It will be shown in a subsequent chapter that a pro
position of moderate complexity has an almost unlimited
number of contradictory propositions, which are more or
less in conflict with the original. The truth of any one or
more of these contradictories establishes the falsity of the
original, but the falsity of the original does not establish the
truth of any one or more of its contradictories, because
there always remains the alternative that nothing is known
concerning the relations of the terms. It may even happen
that no relation at all exists between the terms. In this
view of the matter, then, an assertion of the falsity of a
proposition means its simple deletion. The contrariety is not
between knowledge and knowledge, but between knowledge
and ignorance.
It ought also to be remembered, in dealing with the doc
trine of falsity, that the falsity of all Xs are Ys only
implies that one or more Xs are not Ys. Now in practice
one or a few exceptions are often of no importance ; there
are in many cases singular exceptions which in a sense agree
with, and in a sense falsify, a general proposition. Thus
all points of a revolving sphere describe circles, excepting
xii.] QUESTIONS AND ANSWERS. 119
the two points at the poles. Other examples of singular
exceptions will be found in the Principles of Science, Chapter
XXIX. Professor Henrici points out (Elementary Geometry,
1879, p. 37) that a proposition must be considered to be
true in general, if it be true in an infinite proportion of
cases, and false only in a finite number of exceptions.
This subject of the truth and falsity of propositions as
premises and conclusion may be pursued in Karslake s
Logic, Vol. I. p. 83 ; Whately, Book II. Chap. iii. 2 ;
Aristotle, Prior Analytics, Book II. Chaps, i. iv. ; Port
Royal Logic, Part II. Chap. vii. Watts Logic, Part II.
Chap. ii. 7 and 8.
Most of the scholastic logicians, such as Thomas Aquinas
and Nicephorus Blemmidas, treat this subject elaborately.
9. c Trust (said Lord Mansfield to Sir A.
Campbell) to your own good sense in form
ing your opinions ; but beware of attempting
to state the grounds of your judgments.
The judgment will probably be right ; the
argument will infallibly be wrong.
Explain this phenomenon, and show its logical
significance. [P.]
If you give reasons for a decision, implying that those
reasons are sufficient, and are the reasons upon which you
did make the decision, it is possible for critics subse
quently to inquire whether such reasons logically support
the conclusion derived from them. If they do not, the
judge will be detected in a paralogism which there may be
no means of explaining away. But, if no reasons be given,
120 TRUTH AND FALSITY. [CHAP.
it will seldom be possible for critics to make any such
detection. It is impossible, as a general rule, to publish in
detail the law as well as the evidence upon which a law
case is decided, and, even if it were published, it would
generally be impossible to detect bad logic in a man who
does not assign the precise points on which he relies, and
the way in which he argues about a complex mass of
details.
Although it may be, from his own point of view, con
venient and discreet for a man to avoid giving reasons for
any important public decision, if he can avoid it, yet it is
an open question how far such means of escaping criticism
is likely to increase the carefulness and impartiality of his
judgments. There are many cases, including nearly all the
verdicts given by juries on points of fact, where it would be
highly undesirable to require any statement of reasons.
Where the result depends upon oral testimony, the be
haviour of witnesses, the estimation of degrees of prob
ability and degrees of guilt, it is quite impossible to define
and publish the real premises of the conclusion come to.
We must trust to common sense and judicial tact. The
same remarks may apply to various arbitrations, magisterial
decisions, administrative acts, votes of members of de
liberative bodies. But where the grounds of decision are
precise and brief, so as to be capable of complete state
ment, it seems absurd to suppose that a judge will judge
less well because he needs to disclose his argument. If he
displays bad logic, where bad logic can be judged, he is
clearly not fit to be a judge. Lord Mansfield s advice may
possibly have been prudent and good when given to a man
who was forced to act in novel circumstances, and in a
distant colony (Jamaica), where his decisions would have
more of the nature of administrative acts than law-building
xii.] QUESTIONS AND ANSWERS. 121
judgments. But the decisions of the High Court of
Justice in England not only affect the parties in the cause,
but shape the public law of a large part of the civilised
world, and it is of course requisite that they should be
guided by good logic.
CHAPTER XIII.
EXERCISES REGARDING FORMAL AND MATERIAL TRUTH
AND FALSITY.
1. COMPARE the following syllogisms, or pseudo- syllo
gisms, both as regards their formal correctness, and as
regards the material truth of their premises and conclusion ;
then explain how it is that a materially true conclusion is
obtained in each case.
(1) All existing things are real things ;
No abstract ideas are existing things ;
.-. No abstract ideas are real things.
(2) No real things exist ;
All abstract ideas are real things ;
.-. No abstract ideas are real things.
(3) All real things are existing things ;
No abstract ideas are existing things ;
.-. No abstract ideas are real things.
2. If there be two syllogisms, of which we know that
their major premises are subcontrary propositions, how
may we determine the figure and mood of both ? May
their conclusions be both true in matter ?
3. Prove by means of the syllogistic rules, that given the
CHAP, xiii.] EXERCISES. 123
truth of one premise and the conclusion of a valid syllo
gism, the knowledge thus in our possession is in no case
sufficient to prove the truth of the other premise. [c]
4. It is known concerning a supposed syllogism that it
involves a fallacy of undistributed middle, and that one of
the premises is false in matter; can we or can we not draw
any conclusion under these circumstances ?
5. Construct two syllogisms, such that the major premise
of one shall be the subcontrary of the conclusion of the
other, and such also that the conclusions of both shall be
true in matter. Are these data sufficient to determine the
figure ?
6. If one premise be false in matter, and the syllogism
correct in form, does it follow that the conclusion is false in
matter ?
7. Examine the doctrine that, if the conclusion of a
syllogism be true, the premises may be either true or false ;
but that, if the conclusion be false, one or both of the
premises must be false.
8. Interpret the logical force of the following passage
from Mr. Freeman s Essay on the Holy Roman Empire :
It may have been foolish to believe that the German King
was necessarily Roman Emperor, and that the Roman
Emperor was necessarily Lord of the world.
9. Taking a syllogism of the third figure, and assuming
one of the premises to be false, show whether or not, with
the knowledge of its falsehood thus supposed to be in our
possession, we can frame a new syllogism : if so, point out
the figure and mood to which it will belong.
10. What do you mean by (i) Formal, (2) Material
124 TRUTH AND FALSITY. [CHAP.
truth, as applying (a) to a single proposition, (b) to a
syllogism ?
11. Give a careful answer to the miscellaneous example,
No. 88, in Elementary Lessons in Logic, p. 322.
1 2. Is the following extract sense or nonsense, logically
correct or incorrect ? We may doubt whether the ancient
method of reduction can prove the validity of any syllogistic
mood ; for, as from false premises we can illogically obtain
a true conclusion, the reductio ad impossibile has doubts cast
upon its validity as a method of proof.
13. What is the precise meaning of the assertion that it
is false to say that Castro cannot be proved .not to be
Orton ?
14. If P asserts that oxygen, hydrogen, and nitrogen
cannot be liquefied, and Q denies the assertion, what
precisely must Q be understood to mean ?
15. Analyse all that is implied in the assertion of the
falsity of each of the following propositions :
(1) Roger Bacon was a giant.
(2) Descartes died before Newton was born.
(3) Bare assertion is not necessarily the naked truth.
(4) All kinds of grass except one or two species are not
poisonous.
1 6. Let X, Y, Z, P, Q, R, be six propositions : given
(a) Of X, Y, Z, one and only one is true ;
(b) Of P, Q, R, one and only one is true ;
(c) If X is true, P is" true ;
(d) If Y is true, Q is true ;
(e) If Z is true, R is true ;
EXERCISES. 125
Prove syllogistically, that
(/) If X is false, P is false ;
(g) If Y is false, Q is false ;
(ti) If Z is false, R is false. [c.]
17. How do you meet the following difficulties ?
(1) True premises may by false reasoning give a correct
conclusion ; because, in a train of reasoning there may be
two errors, and one error may neutralise the other.
(2) Since truth applies only to propositions, and a term
per se is incapable of truth, it follows that a term mustier se
always be false, because everything must be either true or
false.
CHAPTER XIV.
PROPOSITIONS AND SYLLOGISMS IN INTENSION.
i. To any one desirous of acquiring a thorough command
of logical science, nothing is so important as a careful study
of the intensive or comprehensive meaning of terms, pro
positions, and syllogisms. This indeed is not an easy task,
as is shown by the fact that some great logicians who have
written upon the subject, especially Sir W. Hamilton, have
fallen into grave errors, or at the best fatal ambiguities of
expression. Most of the common text-books, again, either
ignore the subject altogether, or else treat it in a manner
quite disproportioned to its difficulty and importance. The
following questions and answers touch some of the more
obscure points of the matter, but the student is assumed to
have read the fifth of the Elementary Lessons in Logic, or
else to have studied the subject in one or more of the
following books : Port Royal Logic, Part L, Chapters v. to
vii., Spencer Baynes Translation, 1861, pp. 45 55 (this
work was the first to draw attention to the subject in modern
times); Watts Logic, Part I. Chapter vi., 9 and 10;
Levi Hedge, articles 34 to 38 ; Thomson s Outline of the
Necessary Laws of Thought, 52; Spalding, 1857, 30
to 33 ; Walker s Commentary on Murray s Compendium,
Chapter II. ; Bo wen s Treatise on Logic, Chapter IV.
The most elaborate treatment of the subject is found in
CHAP, xiv.] QUESTIONS AND ANSWERS. 127
Peirce, Proceedings of the American Academy of Aits and
Sciences, 1867, Vol. VII. , pp. 416 432. If the student
read Hamilton s Lectures on Logic, he must carefully observe
what is said about Hamilton in this chapter.
2. State the proposition Men are mortals in
the intensive form.
This proposition, as it stands, is clearly extensive, and
asserts that all individual men will be found among the
things called mortals. When asked to turn such a proposi
tion into the intensive form, students make all kinds of
blunders, saying, for instance, that All the qualities con
noted by the term man are connoted by the term mortal,
or All the properties of men are properties of mortal.
This is certainly not the case, because men, in addition to
being mortal, are rational, are vertebrate, are erect, etc.
Again, a student would say, The attributes of man connote
the attributes of mortality. This means nothing, the verb
connote being wrongly used. To say, again, that All
which possess the properties of man, possess the properties
of mortal, is to leave the proposition just as it was before,
All which possess the properties of man/ being simply
men.
In passing from the extensive to the intensive mode of
thought, there must be a complete inversion of the relation
of the terms. As men are a part of mortals, so the qualities
of mortals are a part of the qualities of men. If we like
we may use a different kind of copula ; but, in that case,
much care is necessary to avoid error. The following are
different modes of expressing correctly the same truth, the
first of each pair being an extensive and the second the
corresponding intensive form of assertion.
128 INTENSION. [CHAP.
c All men are included among mortals ;
-| All qualities of mortals are included among the qualities
v. of men.
J Mortals include men ;
(. Properties of men include the properties of mortals.
J Man is a species of mortal ;
( The genus mortal is in the species man.
( Men are part of mortals ;
( Mortality is part of humanity.
3. Can we exhibit particular and negative propo
sitions in the intensive form ?
This question has not, I think, been much investigated by
logicians, and the remarks to be found in the works of
Hamilton and most other logicians apply only to the
universal affirmative proposition. Taking the particular
affirmative, Some crystals are opaque, it asserts that * One
or more crystals are among opaque things. It follows, no
doubt, that the quality opaqueness is among the qualities
of one or more crystals, namely, the particular crystals
referred to in the extensive proposition. Thus I may be
treated intensively much as A is treated.
Taking the negative proposition, * No iron bars are trans
parent, we cannot infer that No properties of transparent
objects are properties of iron bars. This inference would
be quite false ; for, there may be many properties, such as
gravity, inertia, indestructibility, extension, etc., which are
possessed alike by transparent objects and iron bars. All
we can infer is that Not all the properties of transparent
things are in iron bars, or, Some of the properties of
transparent things are not in iron bars. Entire separation
in extension involves only partial separation in intension, or
xiv.] QUESTIONS AND ANSWERS. 129
an extensive assertion in E gives an intensive assertion in
O. We may also change E into A, getting All iron bars
are non-transparent objects, which of course gives the
intensive form, * The quality of non-transparency is among
the qualities of iron bars.
We may in a somewhat similar way treat the particular
negative, say Some crystals are not symmetrical. We
cannot infer that All the common properties of symmetrical
things are absent from some crystals, but only some of those
properties.
4. How shall we state the following syllogism in
the intensive form ?
All crystals are solids ;
All topazes are crystals ;
Therefore, All topazes are solids.
We must transmute each of the three propositions into
the intensive form, and transpose the premises, thus
All qualities of crystals are qualities of topazes :
All qualities of solids are qualities of crystals ;
Therefore, All qualities of solids are qualities of topazes.
The syllogism is turned, as it were, completely inside
out.
5. Is Hamilton correct in stating the following
as a valid syllogism ?
S comprehends M;
M does not comprehend P ;
Therefore, does not comprehend P.
Professor Francis Bowen (Logic, p. 237) has pointed out
that, as the statement stands (see Hamilton s Lectures, Vol.
K
130 INTENSION. [CHAP.
III. pp. 315, 316, American Edition, p. 223), it is simply
illogical. It involves a fallacy of Illicit Process of the
Major Term ; in short, S may comprehend P through
others means besides M. But Professor Bowen thinks that
Hamilton s error lay in the choice of language, which was
such as no one would understand it as Hamilton really in
tended it. The matter, however, is too important to be
passed over in this way, and I proceed to notice other
places in which Hamilton has treated of intensive
syllogisms.
In his sixteenth Lecture on Logic (Vol. III. p. 295), we
find the following :
An Extensive Syllogism. An Intensive Syllogism.
B is A C is B
C\$ is A
C is A C is A
All man is mortal ; Caius is a man ;
But Caius is a man ; But all man is mortal ;
Therefore, Caius is mortal. Therefore, Caius is mortal.
Between the syllogisms as thus stated there is no difference
whatever, except the transposition of premises, a mere
difference in the order of writing which is immaterial to the
point in question. It is true that Hamilton goes on to
explain as follows :
< In these examples, you are aware, from what has pre
viously been said, that the copula in the two different
quantities is precisely of a counter meaning ; in the quantity
of extension, signifying contained under ; in the quantity of
comprehension, signifying contains in it
Afterwards the example of a concrete syllogism before
given, is thus fully stated in the intensive form (p. 296).
xiv.] QUESTIONS AND ANSWERS. 131
* The Major term Caius contains in it the Middle term
man ;
But the Middle term man contains in it the Minor term
mortal ;
Therefore, the Major term Caius contains in it the Minor
term mortal.
To say the least, this is a very clumsy and misleading
mode of explanation ; for after all, the point of the matter
is left untouched, namely, that it is individual things which
are contained under in the extensive sense, and qualities, or
attributes, which are contained in the other sense. Is it
not absurd to say that Caius contains man, without ex
plaining that man is here taken intensively ? A thing does
not contain any of its qualities in the same way that the
class man, extensively regarded, contains one of its members
or significates, namely, Caius. Nor is the matter much
mended by referring back to the previous explanation
(p. 274), where Hamilton illustrates the relations, saying,
Thus the proposition, God is merciful, viewed in the one
quantity, signifies God is contained under merciful, that is, the
notion God is contained under the notion merciful ; viewed
as in the other, means, God comprehends merciful, that is,
the notion God comprehends in it the notion merciful? This
is again all wrong, unless we interpret notion and containing
in a totally different sense in the second as compared with
the first statement. Even if Hamilton understood the
matter correctly himself, he ought to have stated it unequi
vocally, and not to have left the reader to put the matter
right by careful interpretation of the same words in two
different senses. His subsequent exposition of the sorites
is even worse ; for he gives the identically same premises
twice over in different order, and asserts, without other
explanation, that one is in comprehension and the other in
132 INTENSION. [CHAP.
extension. Suppose it were stated in a police court that
Brown struck Robinson, and then Robinson struck
Brown. Should we not be surprised to learn subsequently
that the verb struck was used in a psychological sense in
one case and a physical one in the other, the real meaning
being that Brown struck Robinson as a very disagreeable
fellow, and then Robinson struck Brown on the head ?
Yet this would not be worse than for Hamilton to state a
syllogism or sorites twice over, with an unimportant change
of order, and then assume that the reader takes one state
ment to be in extension and the other in intension.
I am obliged, therefore, to coincide with the opinion of
De Morgan that Hamilton really missed the point of the
question, in short, did not understand what he was writing
about. The whole matter is put in the clearest light by the
following few lines from De Morgan s Syllabus, pp. 62-3 :
The logicians who have recently introduced the distinc
tion of extension and comprehension, have altogether missed
this opposition of the quantities, and have imagined that
the quantities remain the same. Thus, according to Sir
W. Hamilton, "All X is some Y" is a proposition of com
prehension, but "Some Y is all X" is a proposition of
extension. In this the logicians have abandoned both
Aristotle and the Laws of Thought from which he drew the
few clear words of his dictum : " the genus is said to be part
of the species ; but in another point of view (aAAws) the
species is part of the genus." All animal is in man, notion
in notion : all man is in animal, class in class. In the first,
all the notion animal part of the notion man ; in the second,
all the class man part of the class animal. Here is the
opposition of the quantities.
The same view is more fully stated in De Morgan s Third
Memoir on the Syllogism (pp. 17-19; Camb. Phil. Trans.
xiv.J QUESTIONS AND ANSWERS. 133
1858, pp. 188-9). On the whole, I conclude that Hamil
ton s treatment of the subject is so doubtful and confusing
that it had better not be studied in an elementary course of
reading.
De Morgan, in the paper just referred to, gives some
remarks about the history of the doctrine of intension and
extension, and speaks of Hamilton as a logician who has
recently contended for the revival, or rather the full intro
duction, of the distinction of extension and comprehension.
He correctly names the Port Royal Logic as being the first
modern work to insist on the distinction, though the use
made of it is * not very extensive. But he names only one
other work, the Institutiones Philosophies of J. Bouvier
(.3d Ed. Mans, 1830), as describing this distinction.
De Morgan s reading of modern logic was not extensive.
Not to mention the familiar Watts Logic, in which the
doctrine is frequently dwelt upon (see Part I. Chap. III.
Section 3 ; Chap VI. Section i o, and elsewhere), I
find the matter excellently explained in 1816 in the
brief manual of the American logician, Levi Hedge (pp.
42-44). In Murray s Manual, formerly much used in
Dublin and Glasgow, the subject is fully explained, and
in the clearest possible manner, in the Commentary of John
Walker on Chapter II. This is in fact one of the best
pieces of logical exposition which I know. Walker re
marked that he had treated the point fully, because he
regarded it as absolutely necessary to the understanding of
the subsequent pages, which were often puzzling to students
not familiar with the distinction between the comprehension
and extension of a term. With some regret I must hold,
then, that the pretensions of Hamilton in this matter are
mistaken and unfounded.
The whole subject of extension and comprehension or
134 INTENSION. [CHAP. xiv.
intension has been investigated with much care and pro
fundity of thought by the American logician, Professor
C. S. Peirce, in the memoir already referred to (see p. 127).
This memoir should be studied by those who wish to
acquire a thorough understanding of logical principles and
relations.
6. It is asserted by some logicians that the
predicate of a proposition must be interpreted
in intension while the subject is regarded in
extension. Give your opinion upon this point,
and explain the bearing of the question upon
recent logical controversies. [c.]
I should answer this question to the effect that a propo
sition being, conformably to the opinion of Condillac,
necessarily of the nature of an equation, it is absurd to
suppose that things can be equated to their own qualities
or circumstances. A proposition in extension expresses the
identity of a thing or class of things with the same thing
or class under another designation. As De Tracy says
(Id eologie, Vol. III. p. 529), Dans tout jugement, les deux
idees comparees sont necessairement egales en extension?
A proposition in intension expresses an identity between
the attributes of the one member and those of the other.
The subject may be pursued in my Essay on Pure Logic,
or the Logic of Quality apart from Quantity, 1864, passim ;
in J. S. Mill s Logic, Book I. Chapter V. ; and in Dr.
Martineau s review of Samuel Bailey on the Theory of
Reasoning, in his Essays Philosophical and Theological, 1869,
Vol. II.
CHAPTER XV.
QUESTIONS ON INTENSION.
T. Christian/ animal, Episcopalian, organised,
man. Arrange these terms (i) in the order of exten
sion, beginning with the most extensive; and (2) in the
order of comprehension, beginning with the most com
prehensive. [L.]
2. Arrange the following in the same manner : General,
animal, composer for the pianoforte, Roman, historian of
his own campaigns, conqueror of Gaul. (See De Morgan,
Third Memoir, pp. 20, 21.)
3. Arrange in order of extension and intension such of
the terms given in Question i of Chapter II. as are the
names of subaltern, genera, and species, and can be arranged
in a series.
4. Analyse the following terms in the counter quantities
or wholes of extension and intension : Man, government,
law, triangle, vegetable. [L.]
5. Show that the analysis of an intensive equals the
synthesis of an extensive whole. [c.]
6. Invent a syllogism in Barbara, and state it both in the
extensive and in the intensive forms. [L.]
136 INTENSION. [CHAP. xv.
7. What is the place of the Major and Minor Terms in
the conclusion of (a) an extensive, and (b) an intensive
(comprehensive) syllogism ?
8. Can the distinction of extension and intension be
made to apply to the inductive syllogism? [c]
9. Select from pp. 91 to 98 examples of the moods
Celarent, Cesare, and Camenes, and state them in the
intensive form.
10. What is the difference of meaning of genus and
species in extent and intent ? Is the extent of a notion
always less as the intent is greater, and vice versa ?
1 1. Interpret the following propositions in extension and
intension :
A libel is a malicious and injurious statement.
He who believes himself to be always right in his opinion
claims infallibility.
It is impossible to be and not to be.
He that can swim needs not despair to fly.
CHAPTER XVI.
HYPOTHETICAL, DILEMMATIC, AND OTHER KINDS OF
ARGUMENTS.
i. SOME attempt will be made in the subsequent chapters
on the Elements of Equational Logic to illustrate the
actual and possible variety of assertions and arguments.
But it will be convenient to give here a few examples of
hypothetical and other arguments in the less common forms.
Several subtle questions arising out of the hypothetical
form of assertion are also considered with some care ;
but it has not been thought necessary to treat all the
various forms of disjunctive and dilemmatic arguments
which will be found described in almost identical terms
in numerous text-books.
2. If virtue is voluntary, vice is voluntary ; but
virtue is voluntary ; therefore so is vice, [w.]
A valid Constructive Hypothetical syllogism, equivalent
to the following categorical one in Barbara :
Beings who can be virtuous at will can also be vicious
at will ;
Men can be virtuous at will ;
Therefore, they can be vicious at will.
138 HYPOTHETICAL ARGUMENTS, ETC. [CHAP.
3. Logic is indeed worthy of being cultivated,
if Aristotle is to be regarded as infallible ; but
he is not : Logic, therefore, is not worthy ot
being cultivated. [w.]
Clearly a false hypothetical syllogism. The antecedent
is, if Aristotle is to be regarded as infallible ; this is denied
in the minor premise. In the categorical form the pseudo-
argument might be stated somewhat as follows :
Those who regard Aristotle as infallible must consider
logic worthy of being cultivated ;
We do not regard Aristotle as infallible ;
Therefore, we do not consider logic, etc.
There is Illicit Process of the Major Term.
4. We are bound to set apart one day in seven
for religious duties, if the fourth command
ment is obligatory on us : but we are bound
to set apart one day in seven for religious
duties ; and hence it appears that the fourth
commandment is obligatory on us. [w.]
The antecedent is * if the fourth commandment is obli
gatory ; the consequent is we are bound, etc. ; it is the
consequent which is affirmed, so that the argument involves
the Fallacy of Affirming the Consequent. It may be put
categorically as follows : Those on whom the fourth com
mandment is obligatory are bound, etc. ; we are bound, etc. ;
therefore, we are among those on whom the fourth com
mandment is obligatory. The fallacy is evidently that of
Undistributed Middle, the pseudo-mood being A A A in
the second figure.
xvi.] QUESTIONS AND ANSWERS. 139
5. (i) If the prophecies of the Old Testament
had been written without knowledge of the
events of the time of Christ, (2) they could
not correspond with them exactly ; (3) and if
they had been forged by Christians, (4) the}
would not be preserved and acknowledged by
the Jews : (5) they are preserved and acknow
ledged by the Jews, (6) and they correspond
exactly with the events of the time of Christ :
therefore they were (7) neither written without
knowledge of those events, (8) nor were forged
by Christians. [w.]
The above argument will be found to consist of two
valid destructive hypothetical syllogisms woven together in
statement. Thus (i) and (2) are the antecedent and
consequent of the first syllogism ; (6) is its negative
minor, and (7) is its negative conclusion. The second
syllogism has (3) and (4) for its antecedent and consequent,
(5) for its negative minor premise, and (8) for its con
clusion.
6. In how many ways can you state the sub
stance of the categorical proposition A wolf
let into the sheep-fold will devour the sheep ?
Isaac Watts, in his Essay on the Improvement of the
Mind, has well pointed out the variety of expression which
may be given to the same real assertion. Thus, as equiva
lents for the above proposition, he gives the following :
If you let a wolf into the fold, the sheep will be de
voured : The wolf will devour the sheep, if the sheep-fold
l)e left open : If the fold be not left shut carefully, the
i 4 o HYPOTHETICAL ARGUMENTS, ETC. [CHAP.
wolf will devour the sheep : The sheep will be devoured
by the wolf, if it find the way into the fold open : There
is no defence of the sheep from the wolf, unless it be kept
out of the fold : A slaughter will be made among the
sheep, if the wolf can get into the fold. There are various
modes of hypothetically stating the result contained in the
categorical original.
7. In a strictly logical point of view, ought it to
be offensive to Captain Jones to say of him
If Captain Jones does run away in battle, he
will live to fight another day ?
This question touches deeply, not only the soldierly repu
tation of Captain Jones, but, what is much more important,
the precise import of propositions. It puts forward Captain
Jones as running away in battle, but it puts this forward
only as a hypothesis, the result of which would be his living
to fight another day. It is quite a different matter, what
meaning such a proposition might be taken to imply in
common life ; things are often said in the form of innuendo.
The mere coupling of a man s name with a disreputable
action, even though the action were expressly denied of him,
raises the question, Why was the assertion made at all
unless to bring the terms together in the mind of the hearer?
If in company a gentleman were suddenly to remark There
is not the least reason to believe that Captain Jones did run
away in his last action ; here is a point blank denial of
any ground for believing an assertion to that effect ; yet
every one would construe such a mal-a-propos denial as
evidence of a wish to raise the question, and possibly start
a rumour, which would presently take a disagreeable affirma
tive form. Thus we see that the logic of conversation is
xvi.] QUESTIONS AND ANSWERS. 141
widely different in apparent nature from the strict logic of
science ; not that it is really different in the end, when
thoroughly analysed. But we constantly deal with illogical,
inaccurate, or even untrustworthy persons, so that we can
seldom be sure that an assertion will be construed and
repeated in the form which we originally gave to it. There
is too much truth in the saying of Talleyrand, that words
were given to us to disguise our thoughts.
8. If Brown says to Jones, Because Robinson is
foolish you have no need to be foolish, does
Brown assert categorically that Robinson is
foolish ?
There can be no doubt that, in the logic of common life,
Brown would be understood to make an imputation upon
the wisdom of Robinson, especially if the remark was not
explained by the previous course of the conversation. But
in strict logic it seems very doubtful whether the conjunction
because should be interpreted differently from if, as in
the last question. The fact of Robinson being foolish is
no reason, etc. Foolishness on the part of Robinson is
no reason for you being foolish. A logical copula must
not be understood to assert the physical existence and
occurrence of its subject or predicate; it only asserts a
relation between them.
9- If P is 2, and Q is R, it follows that P is R :
but suppose it to be discovered that no such
thing as Q exists, how is the truth of the
conclusion, P is R, affected by this discovery ?
I do not see how there is in deductive logic any question
about existence. The inference is to the effect that if the
142 HYPOTHETICAL ARGUMENTS, ETC. [CHAP.
propositions P is Q and Q is R are true, then the conclusion
P is R is true. The non-existence of Q may possibly
render one or both premises materially false, in which case
the reasoning vanishes, but is not logically defective. If
I argue, for instance, that satyrs are creatures half man and
half goat ; and creatures half man and half goat are very
hideous, therefore satyrs are very hideous ; the reasoning is
equally good whether satyrs exist or not. We cannot, of
course, say that the conclusion is materally true, if there
be no objects to which the material truth can apply. But
if I argue that satyrs are creatures half man and half goat,
and such creatures exist in Thessaly, therefore, satyrs exist
in Thessaly ; in this case the non-existence of the middle
term would affect the material truth of the second premise,
and, if this be held false, we cannot affirm the material
truth of the conclusion.
I ought to add that De Morgan in more then one place
assumes that the middle term must have existence, or even
objective existence; thus he says (Syllabus, p. 67): In all
syllogisms the existence of the middle term is a datum] etc.
This is one of the few points in which it is possible to
suspect him of unsoundness.
The student may refer to Hamilton s Lectures, Vol. III.
pp. 454-5, and p. 459, on Sophisms of Unreal Middle;
see also Whately s Analytical Outline, 3.
10. Lias lies above red sandstone; red sand
stone lies above coal ; therefore lias lies above
coal. [W.]
This is one of many examples to be found in the logic
books of arguments which simulate the syllogistic form.
It is often said that they can be solved syllogistically ; but
QUESTIONS AND ANSWERS. 143
certainly this cannot be done by the ordinary rules and
processes of the syllogism. The most that we can get,
even by substitution, is that Lias lies above what lies above
coal. The fact is that the argument is really a mathematical
one, involving simple equations. It is precisely similar to
one which has been thus treated by Professor F. W. Newman
(Miscellanies, 1869, p. 28), and Mr. J. J. Murphy. The
former of these logicians, as quoted by the latter, remarks
* The argument Lead is heavier than silver ; Gold is heavier
than Lead : therefore Gold is heavier than Silver^ brings to
the mind conviction as direct as the simplest of syllogisms.
To say that its validity depends on its being reducible to
syllogism, is wholly unplausible : for to effect the reduction,
you have to make changes of form at least as hard to
accept as the direct argument : and when you have got your
syllogisms, they are more complicated and cumbrous than
the argument as it stands.
Mr. Murphy (The Relation of Logic to Language : Belfast
Natural History and Philosophical Society, i yth February
1875) treats the argument simply as a question of quantity,
thus
Call the weights of gold, lead, and silver respectively
x, y, and z : then x = y + /
y = * + q
x = z + q + /.
In the old logic, the foregoing conclusion could be drawn
only by means of the following syllogism :
That which is greater than the greater is greater than the
less :
The weight of gold is greater than that of lead, and the
weight of lead greater than that of silver :
1 Newman has inadvertently written Lead is heavier than Gold, which
is wrong as to fact.
144 HYPOTHETICAL ARGUMENTS, ETC. [CHAP. xvi.
Therefore the weight of gold is greater than that of
silver.
Considered as fact all this of course is true, but considered
as logic it is wrongly stated. That which is here stated as
the major premise is really the syllogistic canon. It is not
merely a general truth, like the truth that all matter gravi
tates, but a logical principle, lying as near to the first
principles of the science as the axiom that a part of a part
is a part of the whole.
We have only to assume x to be the height of lias, y the
height of red sandstone, and z of coal above any one fixed
datum line, and the same equations represent the argument
at the head of this section.
It may be added that Reid was doubtless right in denying
that we argue syllogistically when we infer that because A
and C are both equal to B they are equal to each other.
We may throw it into the form, Things equal to the same are
equal to each other ; A and C are things equal to the same,
therefore they are equal to each other. But this is a delu
sive syllogism. The inference is really accomplished in
obtaining the major premise. The inferences of equality
are prior to and simpler than the inferences of logic, and
the attempt of Herlinus and Dasypodius to throw Euclid
into the syllogistic form has been rightly ridiculed, because
it is an attempt to prove the more simple and self-evident
by means of the more complex.
Some remarks on this point will be found in De Morgan s
Second Memoir on the Syllogism, 1850, pp. 50, 51; his
Fourth Memoir, 1860, p. 8, etc.; in Mr. Murphy s paper
quoted above ; and in Hallam s remarkable note to Section
129 of Vol. III., Chapter III., of his Introduction to the
Literature of Europe (ist Ed. p. 288 ; 5th Ed. p. in).
CHAPTER XVII.
EXERCISES IN HYPOTHETICAL ARGUMENTS.
i. (i) IF he is well, he will come : he is not well : there
fore he will not come.
(2) If he is well, he will come : he will come : therefore
he is well. [H.]
(3) I am sure he will not come, for he is not well ; and if
well he would come.
(4) He will write if he is well ; but as he is not well,
therefore he will not write.
Analyse the above arguments and point out which are
fallacious, and why.
2. Into how many forms of expression can you throw the
matter of this proposition ? Sulphuric acid combined with
calcium produces gypsum.
3. Throw into the form of hypothetical propositions the
following disjunctives
(1) Either the Claimant is Ortoh, or many witnesses are
mistaken.
(2) The tooth of a mammalian is either an incisor, canine,
bicuspid, or molar tooth.
L
146 HYPOTHETICAL ARGUMENTS. [CHAP.
4. Under which of the commonly recognised forms of
syllogism would you bring the following ?
If A is-ff, Cis>;
If Cis >, is F ;
Therefore, If A is B, E is F. [c.]
5. Are hypothetical propositions capable of conversion ?
If so, convert these
(1) If it has thundered it has lightened.
(2) Unless it has lightened it has not thundered.
6. Which of the following arguments are logically
correct ?
(1) A is , if it is C ; it is not C, therefore it is not >.
(2) A is not B unless it is C ; as it is not C, it is not B.
(3) If A is not B, C is not D; but as A is B, it follows
that C is D.
(4) A is not B, if C is D / C then is not D, for A is B.
7. If the Hypothetical Modus Ponens and Modus Tollens
are taken as corresponding to the Categorical First and
Second Figures, and their typical forms to the Moods
Barbara and Camestres, respectively, what other forms of
the respective Hypothetical Modi would correspond to the
other moods of the respective Categorical Figures ? [R.]
8. If A is true, B is true ; if B is true, C is true ; it C is
true, D is true. What is the effect upon the other assertions
of supposing successively that (i) D is false; (2) that C is
false ; (3) that B is false ; (4) that A is false ?
9. Analyse the following arguments and estimate their
validity.
xvii.] EXERCISES. 147
(1) I shall see you if you do not go; but as you are
going I shall not.
(2) The Penge convicts were guilty of murder, if, after
long continued neglect at their hands, Harriet
Staunton died.
(3) Since the virtuous alone are happy, he must be
virtuous if he is happy, and he must be happy if
he is virtuous.
(4) If there were no dew the weather would be foul : but
there is dew ; therefore the weather will be fine.
[a]
(5) If there are sharpers in the company we ought not to
gamble ; but there are no sharpers in the com
pany ; therefore we ought to gamble. [E.]
(6) I could then only be accused with justice of acting
contrary to my law, if I maintained that Mursena
purchased the votes, and was justified in doing so.
But I maintain that he did not buy the votes,
therefore, I do nothing contrary to the law.
Cicero, Pro L. Mursena, c. iii. (See Devey s Logic,
1854, p. 133.)
10. State in the form of a disjunctive argument the matter
of the First Book of Samuel, chapter xvi. verses 6-13.
1 1. Examine the question whether hypothetical and dis
junctive arguments are reducible to the forms of the
categorical syllogism.
12. Dilemmatic arguments are more often fallacious than
not. Why is this ? [c.]
13. Investigate the logical position of the parties to the
following colloquy from Clarissa Harlowe : Morden
But if you have the value for my cousin that you say you
148 HYPOTHETICAL ARGUMENTS. [CHAP. xvn.
have, you must needs think . Lovelace You must allow
me, sir, to interrupt you. If I have the value I say I have !
I hope, sir, when I say I have that value, there is no cause
for that if t as you pronounced it with an emphasis. Morden
Had you heard me out, Mr. Lovelace, you would have
found that my if was rather an if of inference than of
doubt.
This passage is quoted and discussed by Professor Groom
Robertson in Mind, 1877, Vol. II. pp. 264-6.
CHAPTER XVIII.
THE QUANTIFICATION OF THE PREDICATE.
i. As explained in the preface, I have thought it well to
discuss and illustrate in this book of exercises, the forms of
logical expression and inference recognised by Dr. Thomson
and Sir W. Hamilton. These correspond in most cases
with what De Morgan represented under different systems of
notation. They also correspond to some of the expressions
and arguments current in ordinary life. Although in a scien
tific point of view it is far better to eliminate the logical will-of-
the-wisp some, yet the student is obliged to make himself
acquainted with the pitfalls into which it is likely to lead him.
It is assumed that the reader has studied the brief
account of the Quantification of the Predicate given in the
22nd of the Elementary Lessons in Logic, and he is re
commended to read, on the same subject, either Thomson s
Outline, or else Bowen s account of Hamilton s Logic
(Bowen s Logic, Chapter VIII. ). The study of De Morgan s
and Hamilton s own writings is a more arduous and
hazardous undertaking.
The following are the eight kinds of propositions re
cognised by Hamilton, as described by Dr. Thomson.
Sign. Affirmative.
U All X is all Y.
I Some X is some Y.
A All X is some Y.
Y Some X is all K
Negative. Sign.
No X is Y. E
Some X is not some Y. o>
No X is some Y. -q
Some X is no Y. O
150 QUANTIFIED PREDICATE. [CHAP.
2. Indicate by the technical symbols the quan
tity and the quality of the following propo
sitions :
(1) All primary forces are attractive.
(2) All vital actions come under the law
of habit, and none but vital actions
do.
(3) The best part of every man s education
is that which he gives himself.
(4) Only ungulate animals have horns.
(5) Mere readers are very often the most
idle of human beings.
(6) Most water -breathing vegetables arc
flowerless. [p.]
(1) Is clearly a universal affirmative (A).
(2) As regards its first part is also A; but the exclusive
addition, None but vital actions do, means that,
all not vital actions do not. The two parts
together yield a proposition in U, all vital actions
are all that come under the law of habit.
(3) The best part, being a superlative, is a singular
term, and so is the predicate that part which, etc.
Hence the proposition is an identity in U.
(4) An exclusive proposition equivalent to all not un
gulate animals have no horns, which is the con-
trapositive of, and equivalent to, all horned
animals are ungulate.
(5) Means that a great many mere readers are, etc., and
is in the form I.
(6) Is also a particular affirmative proposition.
xviii.] QUESTIONS AND ANSWERS. 151
3. Does not the proposition Y of Thomson imply
O, that is to say, does not some P is all Q
imply that some P is not Q ?
This seems very plausible, because if some P makes up
the whole of Q, there is, so to say, no room left in <2 s
sphere for any more Ps, the remainder of which must there
fore be not Q. This argument, however, overlooks the fact
that the some P in question may possibly be the whole of
P, so that there may be no remainder excluded from Q.
4. Is the proposition Some men are animals
true ? [E.]
The proposition is true or untrue materially according to
the sense we put upon this troublesome word some. If
we take it to mean one or more it may be all, the pro
position is true in fact, but of course states less than is
known to every one.
We must carefully distinguish between the strict and
necessary logical interpretation of some, and that which
applies in colloquy. De Morgan says (Formal Logic, p. 4),
In common conversation the affirmation of a part is meant
to imply the denial of the remainder. Thus, by " some of
the apples are ripe," it is always intended to signify that
" some are not ripe." There is no difficulty in providing in
formal logic for this use of the word by stating explicitly
the two propositions which are colloquially merged into
one. Thus some of the apples are ripe is really I + O.
5. What results would follow if we were to in
terpret * some As are Bs as implying that
* some other As are not Bs ?
152 QUANTIFIED PREDICATE. [CHAP.
The proposition some As are s is in the form I, and
according to the table of opposition (p. 31) I is true if A is
true ; but A is the contradictory of O, which would be the
form of some other As are not .Z?s. Under such cir
cumstances A could never be true at all, because its truth
would involve the truth of its own contradictory, which is
absurd.
Briefly If A is true, I is true ; and if I implies O, then
A implies the truth of its own contradictory O.
Several logicians have come to grief over this troublesome
word, notably Sir W. Hamilton, who in holding that some
is formally exclusive of all and none, throws all logical
systems into confusion. Woolley commits the same great
mistake in saying (p. 77), In every particular proposition,
therefore, the affirmative and negative mutually imply each
other : if only some A is B, then some A is not -B, and
vice versa?
6. Explain the precise meaning of the propo
sition * Some Xs are not some Ys (the
proposition of Thomson). What is its
contradictory ? Give your opinion of its
importance.
This is one of the eight forms of proposition which
Hamilton, in pursuance of the thoroughgoing quantification
of the predicate, introduced into his system. Now, if
some Y J means any some K, that is to say, if the some
is undetermined and may be any where in the sphere of Y,
this proposition does not differ from some Xis not any Y,
which is the proposition O of the old Aristotelian Logic.
But if some Y is a determinate part of the class Y, less
than the whole, then the proposition becomes a mere empty
xviii.] QUESTIONS AND ANSWERS. 153
truism ; for, however X and Y may be related, some part of
X will be different from some part of Y. Thus all equi
lateral triangles are all equiangular triangles, yet some equi
lateral triangles are not some equiangular ones. If all John
Jones sons are Rugby boys, yet some of John Jones sons
are not some Rugby boys. We see that this proposition w
is consistent with all the other propositions of the system,
in all cases, as De Morgan remarks (Syllabus, p. 24), in
which either X or Y has two or more instances in existence :
its contrary is " X and Y are singular and identical ; there
is but one X, there is but one F, and X is Y." A system
which offers an assertion and denial which cannot be con
tradicted in the same system carries its own condemnation
with it, as well observed by De Morgan.
Archbishop Thomson also rejected this form of propo
sition. He says : If I define the composition of common
salt by saying, " common salt is chloride of sodium," I
cannot prevent another saying that " some common salt is
not some chloride of sodium," because he may mean that
the common salt in this salt-cellar is not the chloride of
sodium in that. A judgment of this kind is spurious upon
two grounds : it denies nothing, because it does not prevent
any of the modes of affirmation ; it decides nothing, inas
much as its truth is presupposed with reference to any pair
of conceptions whatever. (Outline of the Laws of Thought,
1860, 79, p. 137.)
SPALDING, pp. 83, 97-102, etc., symbolises the propo
sition (o, by \ O.
In an examination, candidates almost invariably say that
all Xs are Fs, or all Xs are all Fs, is the contradictory of
some Xs are not some Fs ; and De Morgan (1863, p. 4)
speaks of an unnamed logical author who spoiled his work
with a like blunder.
154 QUANTIFIED PREDICATE. [CHAP.
The chief interest of this proposition u> arises from its
important bearing upon the value of Hamilton s System of
Logic, and his position as a logician. Hamilton insisted
upon the thoroughgoing quantification of the predicate, which
means the recognition and employment of all the eight pro
positions which the introduction of the quantified predicate
renders conceivable. Thus was the key-stone to be put
into the arch of the Aristotelic logic. But if, as Thomson
and De Morgan seem to me to have conclusively shown,
this proposition, w, is valueless and absurd, the key-stone
crumbles and the arch collapses. The same ruin does not
overtake De Morgan s system, because his eight propositions
are not all the same as those of Hamilton ; nor does it
affect in any appreciable degree the views of Thomson and
George Bentham, who did not insist upon the thoroughgoing
quantification of the predicate.
De Morgan has admirably expressed the inherent ambi
guity of this word. He says (Fifth Memoir on the Syllogism,
1863, p. 4), " He has got some apples" is very clear:
ask the meaning of "he has not got some apples," in a
company of educated men, and the apples will be those of
discord. Some will think that he may have one apple ;
some that he has no apple at all; some that he has not got
some particular apples or species of apples.
The subject of particular propositions may be pursued in
Spalding s Logic, 1857, p. 172, and elsewhere; Shedden s
Logic; Hughling s Logic of Names, 1869, p. 31; Thomson s
Outline, fifth edition, section 7 7 ; Hamilton s Lectures, vol.
iv., pp. 254, 279; Devey, 1854, pp. 90-94; De Morgan,
1863, Fifth Memoir.
Mr. A. J. Ellis is particularly exact in his treatment of
this question in his articles in the Educational Times,
1878.
xvm.] QUESTIONS AND ANSWERS. 155
7. Solly says (p. 73) If the premises are "some
B is A, some C is not B" the reason may
logically deduce that some C is not some A.
But this conclusion is not in one of the four
legitimate forms. Is the argument valid in
the quantified syllogism, and if so, in what
mood ?
The propositions are as follow :
Some B is some A I
Some C is not (any B) O
Some C is not some A w
The middle term is distributed once in the minor premise,
and, as both terms of o> are particular, there can be no illicit
process. One premise is negative, and so is the conclusion.
No rule of the syllogism is broken, and the argument is
therefore valid. It appears as I O w in the sixth mood of
the first figure of Thomson s table.
8. Which of the following conjunctions of pro
positions make valid syllogisms ? In the case
of those which you regard as invalid, give your
reasons for so treating them.
First Figure. Second Figure. Third Figure.
EYO UOto AtoQ
AEE rj \J O YEO
I U , [c.]
The pseudo-mood A E E in the first figure gives illicit
process of the major term, because the conclusion E distri
butes its predicate, and the major premise A does not.
The pseudo-mood I U t\ draws a negative conclusion, rj,
156 QUANTIFIED PREDICATE. [CHAP.
from two affirmative premises, but is by oversight given in
Thomson s Table of Modes, figure i, mode xii., second
negative form. It is an obvious misprint for I E ^. (Out
line of the Laws of Thought \ section 103, 5th ed., p. 188.)
In the table as reprinted in the Elementary Lessons in Logic,
p. 1 88 (accidentally the same page as in Thomson!), the
error was corrected in the fifth and later editions. It was
pointed out to me by Mr. A. J. Ellis.
E Y O is valid in the first figure.
In the second figure U O w breaks no rule, but the con
clusion instead of being w (some Xs are not some Zs),
might have been in the stronger form O (some Xs are not
any Zs). The moods U O O and U w w appear in
Thomson s table, column 4, though U O w does not. The
mood t] U O is valid.
In the third figure A w O is subject to illicit process of
the major term, since the conclusion O distributes its predi
cate, which is the undistributed predicate of A in the major
premise. Y E O is not subject to the same objection,
because Y distributes its predicate ; but, in this last case,
the conclusion is weakened, and might have been E; hence
Y E E appears in Thomson s table, tenth mood of third
figure, and Y E O does not. appear.
9. In what mood is the following argument :
Aliquod trilaterum est sequiangulum ; omne
triangulum est (omne) trilaterum ; ergo, ali-
quod triangulum est sequiangulum ?
The first premise, some trilateral figure is an equiangular
figure, is plainly a proposition in I ; the second, all
triangles are all trilateral figures, is as plainly a doubly
universal proposition in U; the conclusion, some triangular
QUESTIONS AND ANSWERS. 157
figure is equiangular, is in I. The middle term, trilaterum.
is distributed in the minor premise, though not in the major;
there is no illicit process, nor other breach of the syllogistic
rules, so that the argument is a valid syllogism in the mood
I U I of the first figure. It appears as the twelfth mood
in the first column of Thomson s table of moods. See,
however, Baynes New Analytic, 1850, pp. 126-7, whence
this example is taken.
10. Does the following argument fall into any
valid mood of the syllogism ?
Some man is all lawyer ;
Any lawyer is not any stone ;
therefore, Some man (i.e. lawyer) is not any stone
(i.e. all the rest are stone).
This example is taken by De Morgan (1863, p. 10) as a
case of Hamilton s mood IV. b, as stated in his Lectures on
Logic, Vol. IV., p. 287, thus, A term parti-totally co-
inclusive, and a term totally co-exclusive, of a third, are
parti-totally co-exclusive of each other. It was called by
De Morgan the Gorgon Syllogism, alluding, I presume, to
the petrifying effect it produces upon all mankind who are
not lawyers. It is plainly in the mood E Y w, and though
it does not appear in Thomson s table, may be considered
a weakened form of E Y O, the seventh negative mood
of the first figure. The point of the matter, however, is that
Hamilton, in his later writings, proposed to depart from
the Aristotelian sense of the mark of particular quantity
some. As stated in his Lectures, Vol. IV. p. 281, the view
which he wished to introduce is that some should mean
some at most, some only, some not all. But, if we
apply this meaning of some to the conclusion of the
158 QUANTIFIED PREDICATE. [CHAP. xvm.
Gorgon Syllogism, it produces the ridiculous result that,
though lawyers are not stone, all the rest of mankind are stone.
De Morgan is unquestionably correct, and this Gorgon
Syllogism brings to ruin Hamilton s long adequately tested
and matured system.
The particulars of the discussion between De Morgan
and Professor Spencer Baynes about this Gorgon Syllogism,
and kindred matters, may be found in the Athenceum of
1 86 1 and 1862 and elsewhere.
It is curious that De Morgan states the Gorgon Syllo
gism differently in the Athenaeum of 2d November, 1861,
p. 582, and in his Fifth Memoir on the Syllogism, p. 10 ;
but the difference is not material to the final issue.
II. The month of May has no "R" in its name ;
nor has June, July, or August : all the hottest
months are May, June, July, and August :
therefore, all the hottest months are without
an " R " in their names.
This is Whately s example No. 117, and as he refers the
student to Book IV., Chap. I., i, which treats of induction,
he evidently regards it as an Inductive Syllogism. It would
have been referred by Hamilton to the Thomsonian mood
E U E, the minor premise being treated as a doubly uni
versal proposition. There can be no doubt, however, that
the minor is really disjunctive, thus : A hottest month is
either May, or June, or July, or August. The major is a
compound sentence, comprising four separate propositions,
May has no R in its name/ June has no R, etc. (See
Elementary Lessons, Lesson XXV., p. 215.)
CHAPTER XIX.
EXERCISES ON THE QUANTIFICATION OF THE PREDICATE.
i. EXPRESS carefully, in full logical form, with quantified
subjects and predicates, the following propositions ; assign
the Thomsonian symbol in each case :
(1) Thoughts tending to ambition, they do plot unlikely
wonders.
(2) Fools are more hard to conquer than persuade.
(3) Heaven has to all allotted, soon or late,
Some happy revolution of their fate.
(4) Justice is expediency.
(5) This is certainly the man I saw yesterday.
(6) Man is the only animal with ears that cannot move
them.
(7) Wisdom is the habitual employment of a patient and
comprehensive understanding in combining various
and remote means to promote the happiness of
mankind.
(8) It is among plants that we must place all the Dia-
tomaceae.
(9) When the age is in the wit is out.
"(10) Every man at forty is either a fool or a physician,
(n) Some men at forty are neither fools nor physicians.
(12) Some men at forty are both fools and physicians.
160 QUANTIFIED PREDICATE. [CHAP.
(13) L fitat c est moi, as Louis the Fourteenth used to say.
(14) There are no coins excepting those made of metal,
if we overlook a few composed of porcelain, glass,
or leather.
(15) Antisthenes said Stlv Kraa-Bat vow rj fipo^ov.
(16) All animals which have a language have a voice, but
not all which have a voice have a language.
(17) The elephant alone among mammals has a pro
boscis.
(18) Prudence is that virtue by which we discern what is
proper to be done under the various circumstances
of time and place.
(19) Whatever is, is right.
(20) There are arguments and arguments.
(21) A dispute is an oral controversy, and a controversy
is a written dispute.
(22) There beth workys of actyf lyf othere gostiy othere
bodily.
(23) The only Roman who gave us a summary of
Aristotle was the only Roman who gave us a
summary of Euclid.
(24) Zenobia declared that the last moment of her reign
and of her life should be the same.
(25) As it asketh some knowledge to demand a question
not impertinent, so it requireth some sense to
make a wish not absurd.
(26) Mankind consists of dark men and fair men.
(27) To say that Mr. Raffles was excited was only
another way of saying that it was evening.
(28) Though all well educated men are not discoverers,
all discoverers are well educated men.
(29) No man is esteemed for gay garments but by fools
and women.
EXERCISES. 161
(3) Quand celui qui ecoute n entend rien, et quand celui
qui parle n entend plus, c est la metaphysique.
(31) Friendship finds men equal or makes them so.
(32) I can fly or I can run.
(33) A man is an ill husband of his honour that entereth
into an action, the failing wherein may disgrace
him more than the carrying of it through can
honour him.
(34) Scribendi recte sapere est principium et fons.
(35) Tools are only simple machines, and machines are
only complicated tools.
(36) The wise man knows the fool, but the fool knows
not the wise man.
(37) It is scandalous that he who sweetens his drink by
the gift of the bees, should by vice embitter
Reason, the gift of the Gods.
(38) A and B and C and D, etc., etc., wear black coats
on Sundays ; in fact every man I know does so.
(39) All the Apostles were Jews, because this is true of
Peter, James, John, and every other Apostle.
(40) A dose of arsenic is given to a living healthy dog.
Soon after the dog dies. Arsenic is therefore a
poison.
2. How can a chain of reasoning, founded on circum
stantial evidence, be represented in syllogistic form ? [E.]
3. Having special regard to the logical sense of some,
what do you think of the validity of the following argument
(Thomson s Syllogistic Mood A E rj) ?
All Fis some X;
No Zis any K;
Therefore, No Z is some X.
M
162 QUANTIFIED PREDICATE. [CHAP.
4. We have been assured that "all X is some Y" is
contradicted by "all Y is some X" a proposition which
cannot be made good except by some being declared not
all (De Morgan, Third Memoir, 1858, p. 24.) Investi
gate this point.
5. Take stone and solid as subject and predicate,
and convince yourself that the proposition in o>, some
stone is not some solid, cannot be contradicted by any
propositions of the forms U, A, I, Y, E, O, >?, having the
same subject and predicate.
6. Write out the various judgments, including U and Y,
which are logically opposed to the judgment, No puns are
admissible. State in the case of each judgment thus formed
what is the kind of opposition in which it stands to the
original judgment, and also the kind of opposition between
each pair of the new judgments. [c.]
7. The judgment, "No birds are some animals," is never
actually made because it has the semblance only, and not
the power of a denial. Examine this statement. [p.]
8. Draw inferences from the following :
If Sir Thomas was imbecile, then Oliver was right ; and
unless Sir Thomas was imbecile, Oliver was not
wrong. [p.]
9. Examine the following arguments in those which are
false point out the nature and name of the fallacy ; arrange
those which are valid syllogisms in the usual form, and give
the symbolic description of the mood.
(i) All the householders in the kingdom, except women,
are legally electors, and all the male householders
are precisely those men who pay poor-rates ; it
follows that all men who pay poor-rates are
electors.
xix.] EXERCISES. 163
(2) All the times when the moon comes between the
earth and the sun, are the sole cases of a solar
eclipse ; the 1 1 th of February is not such a time ;
therefore, the nth of February will exhibit no
eclipse of the sun. [THOMSON.]
(3) All men are mortals, and all mortals are all those
who are sure to die ; therefore, all men are all
those who are sure to die.
(4) The Claimant is unquestionably Arthur Orton : for
he is Castro who is the same person as Arthur
Orton.
10. Which of the following moods are legitimate, and
in what figures : E Y O, Y A A, Y A Y, I Y I, Y Y Y,
A E E ? [M.]
1 1. Examine the validity of the following moods :
Figure I. Figure II. Figure III.
U AU AAA YEE
YOO AYY OYO [c.]
12. Exemplify any of the following moods, and deter
mine in how many figures each is valid : U U U, I U I,
Y U Y, i\ U % 01 U co.
CHAPTER XX.
EXAMPLES OF ARGUMENTS AND FALLACIES.
THIS chapter contains a large collection of examples of
Arguments and Fallacies collected from many sources. They
form additional illustrations and exercises to supplement
what are given in the previous chapters. The student is to
determine in the case of each example whether it contains
a valid or fallacious argument. In the former case he is to
throw the example into a regular form, and assign the
technical description of that form, whether a mood of the
categorical syllogism, or of the hypothetical or disjunctive
syllogism, etc. In some examples two or more syllogisms,
or two or more different forms of reasoning, will be com
plicated together. They must of course be analysed and
exhibited separately.
When the existence of fallacy is suspected, the student
must endeavour to reduce this to a distinct paralogism or
breach of the syllogistic rules, exhibiting the pseudo-mood
or pseudo-form of reasoning. In many cases, however, the
fallacy may be of the kinds described in the Aristotelian
text -books as Semi -logical or Material. These fallacies
have been explained in the Elementary Lessons (Lessons
XX. and XXL), but for convenience of reference a simple
list of the kinds of Fallacies is given below. It has not
been found practicable to undertake in this book a full
CHAP, xx.] ARGUMENTS AND FALLACIES. 165
exemplification of the subject of Fallacies. The student is
therefore referred to the Elementary Lessons named, or to
any of the following writings on the subject :
De Morgan s Formal Logic, Chapter XIII., as amusing
as it is accurate and instructive ; Whately s Logic, Book III.,
perhaps the best and most interesting part of this celebrated
text-book ; Edward Poste s edition of Aristotle on Fallacies.
Paralogisms.
1. Four Terms. Breach of Rule I.
2. Undistributed Middle. Breach of Rule III.
3. Illicit Process of Major or Minor Term. Breach of
Rule IV.
4. Negative premises. Breach of Rule V.
5. Negative Conclusion, from affirmative premises, and
vice versa. Breach of Rule VI.
Breaches of Rules VII. and VIII. can be resolved into
one or other of the above.
Semi-logical Fallacies. Material Fallacies.
1. Equivocation. i. Accident.
2. Amphibology. 2. Converse Fallacy of Accident.
3. Composition. 3. Irrelevant Conclusion.
4. Division. 4. Petitio Principii.
5. Accent. 5. Non Sequitur.
6. Figure of Speech. 6. False Cause.
7. Many Questions.
1. France, having a warm climate, is a wine-producing
country. [E.]
2. Livy describes prodigies in his history ; therefore he
is never to be believed. [E.]
166 ARGUMENTS AND FALLACIES. [CHAP.
3. All the metals conduct heat and electricity ; for
iron, lead, and copper do so, and they are (all) metals.
[E.]
4. A charitable man has no merit in relieving distress,
because he merely does what is pleasing to himself. [E.]
5. What is the result of all this teaching? Every day
you hear of a fraud or forgery, by some one who might
have led an innocent life, if he had never learned to read
or write. [E.]
6. The use of ardent spirits should be prohibited by law,
seeing that it causes misery and crime, which it is one of
the chief ends of law to prevent. [E.]
7. Pious men only are fit to be ministers of religion;
some ignorant men are pious ; therefore ministers of
religion may be ignorant men. [L.]
8. No punishment should be allowed for the sake of the
good that may come of it ; for all punishment is an evil,
and we are not justified in doing evil that good may come
of it. [E.]
9. We know that God exists because the Bible tells us
so ; and we know that whatever the Bible affirms must be
true because it is of Divine origin. [E.]
10. The end of punishment is either the protection of
society or the reformation of the individual. Capital
punishment ought therefore to be abolished. It does not
in fact prevent crimes of violence, and so fails to protect
society, while on the other alternative it is absurd. [E.]
11. The glass is falling; therefore we may look for
rain. [E.]
12. This is a dangerous doctrine, for we find it up
held by men who avow their disbelief in Revelation.
13. If there is a demand for education, compulsion is
unnecessary. [E.]
xx.] EXAMPLES. 167
1 4. Actions that benefit mankind are virtuous ; therefore
it is a virtuous action to till the ground.
15. Slavery is a natural institution ; therefore it is wrong
to abolish it.
1 6. No fool is fit for high place; John is no fool;
therefore John is fit for high place. [E.]
17. He is not a Mahometan, for no Mahometan holds
these opinions. [E.]
1 8. Mind is active ; matter is not mind ; therefore
matter is not active. [E.]
19. He must be a Mahometan, for all Mahometans
hold these opinions. [E.]
20. If we are to believe philosophers, knowledge is
impossible, for one set of them tell us that we can know
nothing of matter, and another that we can know nothing
of mind. [o.]
21. Old age is wiser than youth; therefore we must be
guided by the decisions of our ancestors. [o.]
22. Political assassins ought not to be punished, for
they act according to their consciences. [o.]
23. If education is popular, compulsion is unnecessary;
if unpopular, compulsion will not be tolerated. [o.]
24. Nations are justified in revolting when badly
governed, for every people has a right to good govern
ment. [E.]
25. These two figures are equal to the same figure, and
therefore to each other.
26. Opium produces sleep, for it possesses a soporific
virtue. [E.]
27. Wealth is in proportion to value, value to efforts,
efforts to obstacles : ergo, wealth is in proportion to
obstacles.
1 68 ARGUMENTS AND FALLACIES. [CHAP.
28. When Croesus has the Halys crossed, a mighty army
will be lost.
29. A manor cannot begin at this day, because a court-
baron cannot now be founded.
30. Poeta nascitur, non fit ; how absurd it is then to
teach the making of Latin verses !
31. Aio te ^Eacida, Romanes vincere posse.
32. Every rule has exceptions ; this is a rule, and there
fore has exceptions ; therefore there are some rules that
have no exceptions. [E.]
33. All that perceives is mind; the existence of objects
consists in being perceived ; therefore the existence of
objects necessarily depends on mind. [E.]
34. Some objects of great beauty merely please the eye ;
for instance, many flowers of great beauty, and accordingly
they answer no purpose but to gratify the sight. [H.]
35. A miracle is a violation of the laws of nature; and,
as a firm and unalterable experience has established those
laws, the proof against a miracle, from the very nature of
the fact, is as entire as any argument from experience can
possibly be. [E.]
36. The imagination and affections have a close union
together. The vivacity of the former gives force to the
latter. Hence the prospect of any pleasure with which we
are acquainted affects us more than any other pleasure
which we may own superior, but of whose nature we are
wholly ignorant. [E.]
37. Common salt consists of a metal and a metalloid,
for it consists of sodium and chlorine, of which one is a
metal, and the other a metalloid. [E.]
3 8. The truth is, that luxury produces much good. A
man gives half-a-guinea for a dish of green peas ; how much
gardening does this occasion? (Dr. Johnson.) [o.]
xx.] EXAMPLES. 169
39. Nothing can be produced ; for what exists cannot be
produced, as it is already in existence, and what does not
exist cannot be produced, as, since it is not in existence,
nothing can happen to it. [E.]
40. The earth s position must be fixed, if the fixed stars
are seen at all times in the same situations : now the fixed
stars are not seen at all times in the same situations ;
therefore the earth s position is not fixed. [E.]
41. Protective laws should be abolished, for they are
injurious if they produce scarcity, and they are useless if
they do not. [E.]
42. All who think this man innocent think he should
not be punished; you think he should not be punished,
therefore you think him innocent. [E.]
43. If we are disposed to credit all that is told us, we
must believe in the existence, not only of one, but of two
or three Napoleon Buonapartes ; if we admit nothing but
what is well authenticated, we shall be compelled to doubt
the existence of any. How, then, can we be called
upon to believe in the one Napoleon Buonaparte of
history ? [o.]
44. We cannot know what is false, for knowledge cannot
be deceptive, and what is false is deceptive. [E.]
45. A necessary being cannot be the effect of any cause ;
for, if it were, its existence would depend upon the existence
of its cause, and therefore would not be necessary. [E.]
46. The table we see seems to diminish as we move
from it ; but the real table suffers no change : it was not,
therefore, the table itself, but only its image, that was
present to the mind.
47. The existence of sensations consists in being per
ceived : all objects are really collections of sensations ;
therefore their existence consists in being perceived. [E.]
1 70 ARGUMENTS AND FALLACIES. [CHAP.
48. If the earth were of equal density throughout, it
would be about 2-^ times as dense as water : but it is about
5^ times as dense ; therefore the earth must be of unequal
density.
49. Whatever is conditioned must depend on some cause
external to itself: this world is conditioned by time and
space; therefore this world depends upon some cause
external to itself. [F,]
50. It sometimes happens that an electrical current is
excited, where none but magnetic forces are directly called
into play ; for such a current, in certain cases, is excited in
an electric non-conductor by moving a magnet to or away
from it. [R.]
51. The Quaker asserts that if men were true Christians,
and acted upon their religious principles, there would be no
need of armies. Hence he draws the conclusion that a
military force is useless, and being useless, pernicious.
52. Detention implies at least possession; for detention
is natural possession.
53. Nothing can be conceived without extension : what
is extended must have parts ; and what has parts may be
destroyed. [o.]
54. Had an armistice been beneficial to France and
Germany, it would have been agreed upon by those powers :
but such has not been the case ; it is plain therefore that an
armistice would not have been advantageous to either of
the belligerents. [o.]
55. By the law of nature as soon as Adam was created
he was governor of mankind, for by right of nature it was
due to Adam to be governor of his posterity. [o.]
5 6. When men are pure, laws are useless ; when men are
corrupt, laws are broken.
57. There are many arguments which we recognise as
xx.] EXAMPLES. 171
valid which it is impossible to express in a syllogistic form :
therefore the syllogism is valueless as a test of truth, [o.]
58. No man can be a law to himself; for law implies a
superior who gives the law and an inferior who obeys it ;
but the same person cannot be both ruler and subject, [o.]
59. It is injustice to the intellect of women to refuse
them the suffrage ; for the reigns of many queens, as our
own Elizabeth or Anne, have been famous for literary
productions. [o.]
60. To allow every man unbounded freedom of speech
must be advantageous to the State, for it is highly conducive
to the interests of the community that each individual
should enjoy an unlimited liberty of expressing^ his senti
ments, [o.]
6 1. Your sorrow is fruitless, and will not change the
course of destiny. Very true, and for that very reason I
am sorry. [o.]
62. Because some individuals have in their very child
hood advanced beyond the youthful giddiness and debility
of reason, it only needs a proper system of education to
make other young people wise beyond their years. [o.]
63. Haste makes Waste, and Waste makes Want ; there
fore a man never loses by delay. [o.]
64. If peace at any price is desirable, war is an evil ; and
as war is confessedly an evil, peace at any price is de
sirable, [o.]
65. The two propositions, Aristotle is living, and,
Aristotle is dead, are both intelligible propositions ; they
are both of them true or both of them false, because all
intelligible propositions must be either true or false. [E.]
66. No form of democracy is subject to violent revolu
tions, because it never excludes the mass of the people
from political power. [E.]
172 ARGUMENTS AND FALLACIES. [CHAP.
67. The student of History is compelled to admit the
Law of Progress, for he finds that Society has never stood
still. [ E .]
68. It is fated that I shall or that I shall not recover,
in either of which cases the employment of a physician
is useless, and therefore inexpedient. [E.]
69. The assertion that men much occupied in public
affairs cannot have time for literary occupations is disproved
by such instances as Julius Caesar, Alfred, Lord Bacon, Sir
G. C. Lewis, the Earl of Derby, Mr. Gladstone, and the
late Emperor of the French.
70. Whatever had a beginning in time has limits in
space : the universe has no beginning in time : therefore
the universe has no limits in space.
7 1 . The mollusc is an aggregate of the second order ;
for there is no sign of a multiplicity of like parts in its
embryo.
72. The farmers will not pay in rent more than the net
produce of their farms ; for no trading class will continue a
losing business. [L.]
73. The knowledge of things is more improving than the
knowledge of words. The study of Physics must therefore
be more improving than the study of Languages. [E.]
74. The moral world is far from being so well governed
as the material ; for the former, although it has its laws,
which are invariable, does not observe these laws so con
stantly as the latter. [p.]
75. England has a gold coinage and is a very wealthy
country ; therefore, it may be inferred that other countries
having a gold coinage must be wealthy.
76. Most parents are the best judges of the age at which
their children should be sent to school, and as it is un-
xx.] EXAMPLES. 173
desirable to interfere with those who are the best judges of
their children s interest, it follows that parents should not
be compelled to send their children to school.
77. Among the bodies which do not move in elliptic
orbits are some of the comets ; but all bodies which do
move in elliptic orbits return periodically ; hence, some
bodies which return periodically cannot be comets.
7 8. Some rate-payers are clearly not fit for their duties ;
for all male rate-payers are electors, and some electors who
accept bribes are clearly unfit for their duty of electing
representatives.
79. Whatever is done skilfully appears to be done with
ease ; and art, when it is once matured to habit, vanishes
from observation. We are therefore more powerfully ex
cited to emulation by those who have attained the highest
degree of excellence, and whom therefore we can with the
least reason hope to excel. [L.]
80. It is absurd to maintain that when we cannot avoid
thinking or conceiving a thing, it must be true ; for some
persons cannot be in darkness without thinking of ghosts,
in which they do not believe. [R.]
8 1. How can any one maintain that pain is always an
evil, who admits that remorse involves pain, and yet may
sometimes be a real good ? [c.]
82. The time is past in which the transmission of news
can be measured by the speed of animals or even of
steam ; for the telegraph is not approached by either.
[DE MORGAN.]
83. We enjoy a greater degree of political liberty than
any civilised people on earth, and can therefore have no
excuse for a seditious disposition.
84. Those only who understand other languages are
competent to treat correctly of the principles of their own ;
174 ARGUMENTS AND FALLACIES. [CHAP.
since such a competency requires a philosophical view of
the nature of language in general. [L.]
85. If matter must be merely phenomenal, I must be so
too. [E.]
86. A miracle is incredible, because it contradicts the
laws of nature. [E.]
87. There are no practical principles wherein all men
agree, and therefore none which are innate. [E.]
88. Potash contains a metal; for all alkalies contain a
metal, and potash is an alkali. [E.]
89. Quench not hope; for when hope dies, all dies.
90. That is too bad : you have the impudence to say you
are a materialist, while I know that you are a dancing
master.
9 1 . Blood is a colour ; for it is red, and red is a colour.
92. Every incident in this story is very natural and prob
able ; therefore the story itself is natural and probable.
93. Dolor, si longus, levis ; si gravis, brevis : ergo,
omnino fortiter sustinendus. [L.]
94. Whether we live, we live unto the Lord; and
whether we die, we die unto the Lord : whether we live
therefore, or die, we are the Lord s. [Rom. xiv. 8.]
95. Quand on n a point d amis, on n est pas heureux; les
hommes faux et trompeurs n ont point d amis; ainsi les
hommes faux et trompeurs ne sont pas heureux.
96. Philippi was the city where the first Christian Church
in Europe was founded : it was also the place where the
republican army of Rome under Brutus and Cassius was
finally defeated; hence the republican army was finally
defeated at the city where the first Christian Church in
Europe was founded.
97. Switzerland is a republic, and, you will grant, a more
stable power is not to be found ; nor, again, is any political
EXAMPLES. 175
society more settled than that of the United States. Surely,
then, republican France can be in no danger of revolution.
[R.]
98. Expand into a syllogism, as briefly as you can, the
argument contained in the dialogue of Shakespeare s King
Henry VI. Part III. Act I. Scene i, between the words,
my title s good, etc., and, succeed and reign. [o.]
99. Quoniam deos beatissimos esse constat ; beatus
autem sine virtute nemo potest : nee virtus sine ratione
constare ; nee ratio usquam inesse nisi in hominis figura ;
hominis esse specie deos confitendum est. [H.]
100. Without order there is no living in public society,
because the want thereof is the mother of confusion, where
upon division of necessity followeth ; and out of division,
destruction. [HOOKER, Ecclesiastical Polity, v. 8, s. i.]
i o i. Whatever is contradictory to universal and invariable
experience is antecedently incredible ; and as that sequence
of facts which is called the order of nature is established,
and in accordance with universal experience, miracles or
alleged violations of that order are antecedently improbable.
[E.]
102. Justice is the profit of others; therefore it is un
profitable to the just man to be just. r o 1
103. In trade both buyer and seller profit : in the home
trade both these profits remain in the country; in the
foreign trade one profit goes to the foreign trader ; therefore
the same population will be more profitably employed in
the home than in the foreign trade. [o "I
104. Whatever brings in money enriches. Hence the
value of any branch of trade, or of the trade of the country
altogether, consists in the balance of money it brings in ;
and any trade which carries more money out of the country
than it draws into it is a losing trade : and therefore, money
176 ARGUMENTS AND FALLACIES. [CHAP.
should be attracted into the country, and kept there by
bounties and prohibitions.
105. Distinction may be reasonably expected, because
what is not uncommon may be reasonably expected, and
distinction is not uncommon. [E.]
1 06. Neither am I moved with envy; for if you are
equal to, or less than myself, I have no cause for it ; and if
you be greater, I ought to endeavour to equal you, and not
to speak evil of you. [L.]
107. Great men have been derided, and I am derided :
which proves that my system ought to be adopted.
[DE MORGAN, Paradoxes, p. 387.]
1 08. Preventive measures are always invidious, for when
most successful the necessity for them is the least apparent.
109. Treason never prospers : What s the reason ? Why,
when it prospers, none dare call it Treason.
no. Neque quies gentium sine armis, neque arma sine
stipendiis ; neque stipendia sine tributis habere queunt.
[TACITUS, Hist. Lib. iv. cap. 74.]
in. Men are not brutes; brutes are irrational: all
irrational beings are irresponsible ; therefore, men are not
free from responsibility. [H.]
112. The best of all taxes are taxes on consumption and
taxes on the transfer of property : now all the latter and
many of the former are levied by stamps; stamp duties
therefore are good taxes, and taxes on justice are all stamp
duties ; therefore taxes on justice are good taxes. (See
BENTHAM S Protest against Law Taxes, second edition,
1816, pp. 53, 54-)
113. Dr. Johnson remarked that a man who sold a
penknife was not necessarily an ironmonger. What is the
name and nature of the logical fallacy against which this
remark was directed?
xx.] EXAMPLES. 177
114. When Columbus made the egg stand on end by
breaking it, what fallacy may he be said to have committed?
115. Either all things are ordered by an intelligent Being
who makes the world but one family (and if so, why should
a part, or single member complain of that which is designed
for the benefit of the whole?); or else we are under the mis
rule of atomes, and confusion. Now, take the case which
way you please, there s either no reason or no remedy for
complaint ; and therefore it is to no purpose to be uneasie.
[Marcus Antoninus Meditations, ix. 40.]
1 1 6. Silk is dearer than wool, and wool than cotton ;
therefore silk is dearer than cotton.
117. One napoleon is worth twenty francs, and twenty
francs are worth about sixteen shillings ; therefore one
napoleon is worth about sixteen shillings.
1 1 8. The Prince of Wales is the eldest son of the reign
ing sovereign ; and the eldest son of the reigning sovereign,
if there be such, is the heir to the throne ; therefore the
Prince of Wales is heir to the throne.
119. It is a mistake to suppose that the rents paid to
landlords are a burden on the public, since corn would not
be more plentiful or cheaper if they were abolished.
120. The Greeks have little respect for the petty honesty
of small tradesmen ; we do not greatly admire the wiles of
Ulysses ; therefore any common inward standard of morals
is impossible.
121. Dissent always weakens religion in the people ; for
it sets itself in opposition to the National Church.
122. We are not inclined to ascribe much practical value
to that analysis of the inductive method which Bacon has
given in the second book of the Novum Organum. It is
indeed an elaborate and correct analysis. But it is an
analysis of that which we are all doing from morning to
N
178 ARGUMENTS AND FALLACIES. [CHAP. xx.
night, and which we continue to do even in our dreams.
(MACAULAY, Essay on Bacon], [E.]
123. The Claimant has undoubtedly many peculiarities
of gait and manner which were characteristic of the missing
baronet. Are not these therefore proofs of identity equi
valent to the evidence of imposture afforded by the
absence of tattoo-marks which the genuine man is proved
to have possessed ?
124. Even if it could be shown that animals perform
certain actions which men could only perform by the aid of
reason, it would by no means necessarily follow that animals
perform them by its aid. [c.]
125. If there s neither Mind nor Matter,
Mill s existence, too, we shatter :
If you still believe in Mill,
Believe as well in Mind and Matter. [E.]
126. If we accept Aristotle s testimony, we may infer
that Anaximander was not one of the Ionian philosophers
that accepted as the One material principle a mean term
between Water and Air ; for, in the Physics, we read that
he held the substances in nature to have been produced
from the primordial principle by a process of secretion and
not by a process of condensation and rarefaction ; while in
the De Ccelo it is stated that other mode of production
than the last-named was not put forward by any who
adopted such a mean term for their principle. What
syllogistic form (figure and mood) does this inference most
naturally assume ? [ R ]
CHAPTER XXI.
ELEMENTS OF EQUATIONAL LOGIC.
i. THE symbols employed in this system are the
following :
A, B, C, or other capital letters, signify qualities, or
groups of qualities, forming the common part, or intensive
meaning, of terms, or names of objects and classes of
objects.
a, b, c, or other small italic letters, are the corresponding
negative terms ; thus a signifies the absence of one or more
of the qualities signified by A. This notation for negatives
was proposed by De Morgan (Formal Logic, p. 38). The
mark = is the sign of Identity of Meaning of the terms
between which it stands ; thus A = B indicates that the
qualities signified by A are identical with the qualities
signified by B.
The sign ) signifies unexdusive alternation, including
the ordinary meanings of both the conjunctions or and and.
Thus A [ B means the qualities of A or those of B, or
those of both A and B, if they happen to coincide.
Juxtaposition of two letters forms a term whose meaning
is the sum of the qualities signified by the two letters :
thus AB means a union of the qualities of A and B.
,8o EQUATIONAL LOGIC. [CHAP.
2. The Laws of Combination of these symbols are as
follow :
The Law of Commutation. AB - BA : that is to say,
the sum of qualities of A and B is evidently the same as
the sum of qualities of B and A. The way of arriving at
the sum may be different, but the result is identical.
The Law of Simplicity. AA - A : if we have the
same qualities twice over we get the same as if we named
them once.
The Law of Unity. A -|- A = A : the qualities of A
J the qualities of A are simply the qualities of A.
or
The Law of Distribution. A(B -|- C) - AB -|- AC.
The qualities of A with those of B ~ those of C are the
same as those of AB - - those of AC.
or
The Law of Indifferent Order. B -|- C = C -|- B, which
is sufficiently evident.
3. The Laws of Thought are the foundation of all
reasoning, and may thus be symbolically stated :
The Law of Identity .... A - A.
The Law of Duality or of ) AB ^
Excluded Middle . . . }
The Law of Contradiction . Aa - o.
The successive application of the Law of Duality to two,
three, four, five or more terms, gives rise to the development
of all possible logical combinations, called the Logical
Alphabet, the first few columns of which are given below.
The combinations for six terms are given in the Principles
of Science, p. 94 (first ed. Vol. I. p. 109).
xxi.] ELEMENTS. 181
THE LOGICAL ALPHABET.
i. ii. in. iv. v. vi. vi. continued.
AX AB
ABC
A B C D
ABCDE
BCDE
a X Kb
KB c
AB C d
ABCD*
B CD e
a B
Kb C
A B c D
ABCaTE
B C^E
a b
Kb c
A B c d
AB Cde
aE C d e
/r T-? C*
Kb C D
ABcDE
a E cD E
aE c
A3 C rf
A B cDe
a E c D
ab C
Kb c D
AB cdE
aE c d E
a b c
Kb c d
A B c d e
B c d e
aE C D
A-5CDE
a b C D E
B C rf
KbC De
a* C D e
B c D
KbC dE
ab Cd E
B * rf
K b C d e
ab C d e
a b C D
K b c D E
ab c D E
a b C rf
Kb c D e
c D ,?
a 3 c D
Kb c d E
*Z (y <7 ^/ E
abed
Kb c d e
ab c d e
4. The one sole and all sufficient rule of inference is the
following RULE OF SUBSTITUTION.
FOR ANY TERM SUBSTITUTE WHAT IS STATED IN ANY
PREMISE TO BE IDENTICAL IN MEANING WITH THAT TERM.
The term may consist of any single letter, any juxtaposed
letters, or any group of alternatives connected by the
sign ] , the sign of unexclusive alternation.
5. It is assumed as a necessary law that every term must
have its negative. This was called the Law of Infinity in
my first logical essay (Pure Logic, p. 65 ; see also p. 45); but
as pointed out by Mr. A. J. Ellis, it is assumed by De Morgan,
in his Syllabus, article 1 6. Thence arises what I propose to
call the CRITERION OF CONSISTENCY, stated as follows :
Any two or more propositions are contradictory when, and
only when, after all possible substations are made, they
occasion the total disappearance of any term, positive or
negative, from the Logical Alphabet.
The principle of this criterion was explained in p. 65 of
the Essay on Pure Logic referred to, but subsequent inquiry,
and the writings of Mr. A. J. Ellis, have tended to show
the supreme importance of the criterion.
182 EQUATIONAL LOGIC. [CHAP.
The processes of this equational system of Logic are
fully treated in the first seven chapters of the Principles of
Science, and they are now amply illustrated by the problems
which follow.
6. How do you express in the new logic the four
Aristotelian forms of proposition indicated by
the vowels A, E, I, and O ?
The answer is
A. Every A is B. A = AB (i).
E. No A is B. A - Kb (2).
I. Some A is B. CA = CAB (3).
O. Some A is not B. CA = CA (4).
The first expresses the coincidence of the class A with
part of the class B, namely A B, which is the equational
mode of asserting that the A s form part of the B s. The
second expresses similarly that the A s are found among
the not-B s. In the third form some is expressed by the
symbol C, and the proposition asserts that some A s (C A)
are identical with a part of the class B. Some difficulties
may arise about this form, owing to the ambiguity of the
Aristotelian some, as elsewhere discussed (pp. 151-158).
The fourth proposition is evidently the negative form of
the third.
7. How shall we express equationally the asser
tion of Hobbes (De Corporc Politico, I. i. 13),
that Irresistible might in the state of nature
is right ?
* Might is the principal part of the subject, but it is
qualified or restricted in this proposition by the adjective
XXL] QUESTIONS AND ANSWERS. 183
irresistible, and by the adverbial in the state of nature.
Thus putting
A = irresistible ; C = in the state of nature ;
B = might : D = right.
The 1 subject is clearly ABC; and D is affirmed of it.
But Hobbes cannot, of course, have meant that all right
is irresistible might ; only in the state of nature is this true.
As, indeed, irresistible might must overcome everything
opposed to it, there can be nothing else right in the sphere
of its action, so that the proposition would seem to have
the form A B C = C D. It is not easy to be sure of the
meaning even of Hobbes.
8. Represent the meaning of the sentence Man
that is born of a woman, is of few days, and
full of trouble.
The relative clause, that is bom of a woman, is
evidently explicative, and we cannot suppose that there
are any men not born of a woman. Hence taking
A = man ; C = of few days ;
B = born of a woman ; D = full of trouble ;
the meaning seems to be expressed in the two pro
positions
A = AB;
A = ACD.
But if we may treat the sentence as an imperfectly expressed
syllogism namely, because man is born of a woman he
is, etc. then the premises obviously become A = AB,
and B = B C D, and the conclusion by substitution for B
in the first of its value in the second, is A = ABCD.
1 84 EQUATIONAL LOGIC. [CHAP.
p. Show how to obtain equationally the contra-
positive of A = AB.
This is explained in the Principles of Science, p. 83
(first edition, Vol. I. pp. 97-102), but may be thus briefly
repeated
By the Second Law of Thought
b = Kb -|. ab.
Substitute for A its equal AB.
b = AB -I- ab = o -|- ab,
or, b = ab.
Concerning the contrapositive see above, pp. 32,
43-47, etc.
10. Show how to obtain the complete contra-
positive of A = B.
As before
b = Kb -|- ab = B# ] ab = o -|- ab = ab \
similarly
a aB .|. ab = aK -|- ab = o -| ab = ab.
Having now the two propositions
a ab = b,
it is plain that we may eliminate ab, and get
a = b.
11. What descriptions of the terms glittering
thing and * not gold can you draw from the
following assertions ?
(1) Brass is not gold ;
(2) Brass glitters.
Let A = brass ; B = golden ;
C = glittering thing.
The premises are
(i) A = Kb; (2) A = AC.
XXL] QUESTIONS AND ANSWERS. 185
Obviously C = ABC -|. KbC -|- aEC -|- abC.
The first of the alternatives ABC is negatived by (i) ;
but the second and fourth coalesce, and we have
C = bC -|. tfBC;
that is, a glittering thing is either not golden, or else it is
golden, and then not brass.
For b we similarly get
b = bC -I- abc.
Show that we may also infer
C = C (a -|. b\
and b = b (a -|- C).
12. How shall we represent in the forms of
Equational Logic the moods of the old syllo
gism ?
All of the moods without exception may be solved by
the indirect method, that is by working out the combinations
consistent with the premises. Most of the moods may, how
ever, be solved also by direct substitution, as will be seen
in the following examples :
Barbara.
All men are mortal; (i) B - BC.
All kings are men; (2) A = AB.
. . All kings are mortal ; (3) A = ABC.
We get (3) by substituting for B in (2), its equivalent
BC in (i). The conclusion amounts to saying that king
is equivalent to * king man mortal. If desired, we can
by further substitution of A for A B in (3) obtain A = AC,
or king = king mortal, which is a precise expression for
the Aristotelian conclusion All kings are men.
1 86 EQUATIONAL LOGIC. [CHAP.
Celarent.
No men are perfect ; ( i ) B = B<r.
All kings are men; (2) A = AB.
No kings are perfect ; (3) A = ABr.
Solved, as in the last case, by direct substitution in (2)
of the value of B given in (i).
Darii.
All mathematicians have well-trained ~| _ ~,
intellects ; j
Some women are mathematicians ; AB = ABC.
Some women have well - trained ") A-orr*
c AJJ AijC^U.
intellects. )
Here A stands for the indefinite adjective some, and B
for women, and we then treat AB as an undivided term,
and obtain the result by direct substitution, exactly as in
the previous moods.
Ferio.
No foraminifera are fresh - water \ _
inhabitants ; j
Some components of chalk are fora- |
. . ., C AJj = A-DV_x.
mimfera ; )
Some components of chalk are not ) .
.... > AB = ABG/.
fresh-water inhabitants. j
Except that a negative term d takes the place of the
positive term D, in the last mood, there is no difference in
form between them. In fact, all the four moods of the
first figure present so great similarity that they may be said
to be of one form of inference.
xxi.] QUESTIONS AND ANSWERS. 187
Cesare.
The absolute is not phenomenal ; (i) C = Cb.
All known things are phenomenal ; (2) A = AB.
All known things are not the absolute. (3) A = ABr.
We cannot by any direct substitution obtain the conclusion
from the premises, as B appears in (2) and b in (i). But
we may take the contrapositive of (i) as described before
(p. 184), namely, B = B<r, and substitution in the second
premise is then practicable.
Camestres.
All laws of nature are invariable ; (i) C = CB.
No human customs are invariable; (2) A = Ab.
No human customs are laws of nature. (3) A = Ac.
As in the last mood, we must take the contrapositive of
(i), namely b = b c, and substitute thereby in (2).
13. The equational treatment of the moods Camestres,
Cesare, and Camenes is described also in the Principles of
Science, pp. 84-6 (first edition, Vol. I. pp. 99-101), or in
the Substitution of Similars, pp. 47-49; but the following
is the briefest way of getting the Aristotelian conclusion,
of Camestres, as suggested by Mr. W. H. Brewer, M.A.
Let the premises be
(1) A = AC.
(2) B - "Re.
Multiply together, and we get
AB = ABO- = o.
Thus there is no such thing as AB.
Similarly with Camenes.
i88 EQUATIONAL LOGIC. [CHAP.
14. The remaining moods need only be symbolically
represented. In every case D = some :
Major Premise.
Minor Premise.
Conclusion.
Festino .
. A =
Kb .
CD
= BCD.
CD =
tfBCD
Baroko
. A =
AB .
CD
- CD .
CD =
abCV
Darapti
. B =
AB .
B
= CB .
AB =
CB
Disamis .
. BD =
ABD .
B
= BC .
BD =
ABCD
Datisi .
. B =
AB .
BD
= BCD .
BCD =
ABD
Felapton .
. B =
aE .
B
= BC .
BC =
aE
Bokardo .
. BD -
tfBD .
B
= BC .
BCD =
tfBCD
Ferison .
=
aE .
BD
= BCD .
BCD =
tfBD
Bramantip
. A =
AB .
B
= BC .
ABC =
A
Camenes .
. A =
AB .
B
= Br .
C =
aC
Dimaris .
.AD =
ABD.
B
= BC .
AD -
ABCD
Fesapo
. A -
Kb .
B
= BC .
BC =
<zBC
Fresison
A =
Kb
BD
- BCD .
BCD =
0BCD
15. Exhibit the logical force of the motto adopted
by Sir W. Hamilton
(1) In the world there is nothing great but
Man.
(2) In Man there is nothing great but Mind.
Let A - in the world ;
B = man ;
C = possessing mind ;
D = great.
The conditions may be represented as
(i) A = ABD -I- Kbd.
(a) B = BCD -I-
As it may be understood, though unexpressed, that
XXL] QUESTIONS AND ANSWERS. 189
(3) all men are in the world, and that (4) all possessing
mind are men, we have further
(3) B = AB. (4) C = CB.
The combinations are thus reduced to
ABCD abcD
hJbcd abed.
Observe that, if mind were regarded not as an attribute
but as a physical part of man, we could not treat the
assertion as one of simple logical relation.
16. What is the meaning of the assertion that
All the wheels which come to Croyland are
shod with silver ?
If we take
A = wheel ;
B = coming to Croyland ;
C = shod with silver,
the assertion, as it stands, is evidently in the form
AB = ABC (i).
But it was always understood, no doubt, that this adage
was to be joined in the mind with the tacit premise No
wheels are shod with silver, expressed by
A = Kc (2).
There seems, at first sight, to be contradiction between
these premises ; for (i) speaks of wheels shod with silver
and (2) denies that there are such things. The explanation
is obvious, namely, that there are no such things as wheels
coming to Croyland. Of the four combinations containing A,
igo EQUATIONAL LOGIC. [CHAP.
is negatived by (i), and ABC and AC by (2), so that
the description of A is given thus,
A = Kbc,
or, by substitution of A for Ac,
A = A,
that is, no wheels come to Croyland. This is of course
the inference which the adage was intended to suggest,
Croyland being an ancient abbey lying among the fens of
Cambridgeshire, where in former days no wheeled vehicle
could make its way.
This question illustrates the important logical principle
that all propositions ought, strictly speaking, to be interpreted
hypothetically. We have only to put these premises in the
hypothetical form, and we see that they make a reasonable
destructive hypothetical syllogism thus
If any wheels come to Croyland they are shod with
silver ; but no wheels are shod with silver ; therefore,
no wheels do come to Croyland.
17. Ruminant animals are those which have
cloven feet, and they usually have^ horns ; the
extinct animal which left this footprint had a
cloven foot; therefore it was a ruminant animal
and had horns. Again, as no beasts of prey
are ruminant animals it cannot have been a
beast of prey.
The above problem is given in the Elementary Lessons
(p. 321, No. 78). Taking our symbols thus
A = ruminant ; D = extinct animal ;
B = having cloven feet ; E = beast of prey ;
C = having horns ;
we have clearly A = B (i).
XXL] QUESTIONS AND ANSWERS. 191
The statement that ruminant animals usually^ have horns
may be formalised as
BA = BAG (2),
that is to say, a certain particular portion of the class A,
BA have horns. Next we have
D = DB (3).
Substituting in (3) by (i) we get
D = DA (5);
showing that the extinct animal was a ruminant. But, as
we cannot substitute between (3) and (2), it is erroneous to
assert that it had horns. If by usually we mean in the far
greater number of cases, then there is a considerable prob
ability, but no certainty that it had horns.
Again, we have as an additional premise, that beasts of
prey are not ruminant, or
E = ^E (4),
which, taken with
D = DA (5),
our previous conclusion, gives a syllogism in Cesare, estab
lishing that D is not E, or that the extinct animal cannot
have been a beast of prey. This we might sufficiently prove
symbolically by multiplying the respective members of (4)
and (5) together, giving
DE = #EDA = o.
This shows that inconsistency arises from supposing that
this D can be also E. The same result might be worked
out by combinations, giving D = D<?.
1 The first edition of the Lessons reads always instead of usttally.
i 9 2 EQUATIONAL LOGIC. [CHAP.
18. Take the proposition All crystals are solids/
and ascertain precisely what it affirms, what it
denies, and what it leaves doubtful.
[A. J. ELLIS.]
Taking A = crystal, and B = solid, the proposition is
in the form
A = AB.
The conceivable combinations are four in number,
namely,
AB, A, #B, and ab.
Of these, only Ad is inconsistent with the premise, that is
to say, the premise All crystals are solids denies the
existence of such things as * unsolid crystals. We cannot
strike out AB, because then there would be no such thing
as crystals left ; hence the premise affirms the existence of
solid crystals, in the sense that any other proposition deny
ing that crystals are solid, or that solids may be crystals,
would stand in contradiction to our premise.
Again, we may not strike out a b, because there would
then be no such thing as b, or not-solids. Hence to avoid
contradiction of our premise, there must be such a thing
as c a non-crystal which is not-solid. If we are to hold to
our adopted Criterion of Consistency (p. 181), we must
say that at least one case of ab exists, so that to avoid self-
contradiction, some, that is at least one case of not-crystal,
must be allowed to exist l and to be not-solid. This con
firms the conclusion which we previously obtained by the
Aristotelian logic from the same premise (p. 48). But the
combination 0B may be removed or left without affecting
the truth of the premises, which therefore leave it entirely
in doubt whether not -crystals which are solids exist.
1 Concerning the logical sense of the verb exist, see pp. 141-2.
XXL] QUESTIONS AND ANSWERS. 193
Not -crystals, not -solids must exist, but Not crystals-
solids, may or may not. But if they do not, then crystals
and solids will be coincident classes.
To sum up
A = AB affirms that all As are Bs ;
A = AB affirms that all not-Bs are not-As ;
A = AB affirms that some not-As are not-Bs ;
A = AB denies that all not-As are not-Bs ;
A = AB leaves doubtful whether not-A can be B.
Ip. Are the following propositions equivalent
each to the other ?
(1) All who were there talked sense ;
(2) All who talked nonsense were away.
[DE MORGAN.]
Putting
A = being there ; a = not being there ;
B = talking ; b = not talking ;
C = sensible ; c = not sensible ;
the first premise seems to mean that all who were there
(A), talked and were sensible ; that is
A = ABC (i).
The second is to the effect
B* = a&c (2) ;
that is to say, those who talked (B), but not sensibly
(c\ were away (a). Now (i) negatives three combinations,
AB<r, A<5C, and Kbc\ whereas (2) negatives only AB<r. They
are, therefore, very different propositions; for (2) allows
that some may have been present who did not talk at all
(b\ whether sensible people or not (C or c). Nevertheless,
there is this much relation between the two propositions,
that we can infer (2) from (i). If all who were there talked
o
i 9 4 EQUATIONAL LOGIC. [CHAP.
sense, those people who talked nonsense, assuming there to
be such persons, must have been absent. But we cannot
invert this relation. Because those who talked nonsense
were away, it does not follow that those who were present
talked sense ; they may all have been silent.
20. De Morgan says (Syllabus, p. 14), Any one
who wishes to test himself and his friends upon
the question whether analysis of the forms of
enunciation would be useful or not, may try
himself and them on the following question :
Is either of the following propositions true, and
if either, which ?
(1) All Englishmen who do not take snuff
are to be found among Europeans who
do not use tobacco.
(2) All Englishmen who do not use tobacco
are to be found among Europeans who
do not take snuff.
Required immediate answer and demonstra
tion.
Assigning symbols as follows :
A = Englishmen ; C = taking snuff;
B = Europeans j D = using tobacco ;
it is pretty obvious that the above propositions are thus
symbolised
(1) Ac =
(2) M =
We are to compare these with the well-known relations
of the terms, which may be assumed to be
(3) A = AB ;
xxi.] QUESTIONS AND ANSWERS. 195
that is, * Every Englishman is an European, and
(4) C = CD;
that is, All who take snuff use tobacco. Now, in working
out the combinations, we find that the class Ac is composed
under conditions (3) and (4) as follows :
A^ = ABd3 -I- ABrt
The truth is, then, that Englishmen who do not take snuff
consist of English Europeans not taking snuff, but using
tobacco, and of English Europeans neither taking snuff nor
using tobacco. In short (i) is erroneous in ignoring the
fact that some Englishmen not using snuff may be Europeans
who do use tobacco for smoking.
According to assumption (2) the description of h.d is
AIW, which coincides with the description drawn from
(3) and (4). Thus it is true that all Englishmen who do
not use tobacco are to be found among Europeans who do
not take snuff ; the negation of the larger term, using tobacco,
includes the negation of the narrower one, using snuff.
But it by no means follows that because our inference about
Ad is the same from (2) as from (3) and (4), therefore these
conditions are identical, as will be seen in the following
descriptions of the class A as furnished under the several
suppositions and conditions
(1) A - ABC ! AEcd -I-
(2) A = ABCD i AB<r -I-
(3) and (4) A = ABCD -|- AB*.
21. What can we infer about the term Europeans
from the following premises ?
(1) All Continentals are Europeans ;
(2) All English are Europeans ;
(3) No English are Continentals.
I9 6 EQUATIONAL LOGIC. [CHAP.
Taking A, B, C to represent Continentals, English, and
Europeans respectively, the premises become
(1) A - AC.
(2) B = BC.
(3) B = 0B.
The combinations left uncontradicted are the four
AC, 0BC, abC, abc, whence we learn that Europeans,
C, consist of Continentals who are not English, of English
who are not Continentals, and of any others, who are neither
Continentals nor English (abC).
22. Criticise Thomson s Immediate inference
by the sum of several predicates. . . . From
a sufficient number of judgments in A, having
the same subject, a judgment in U may be
inferred, whose predicate is the sum of all the
other predicates.
This question has been answered in the Principles of
Science, p. 61 (first ed. Vol. I. p. 73). Judgments in A are
of the form P = PQ, P = PR, P = PS, etc, and by
summing up the predicates by successive substitution in the
second side of P - PQ, we may get P == PQRS
But this does not give a proposition of the form U which,
as described by Thomson, is of the form P = X.
23. Represent the following argument from
Thomson s Laws of Thought, 107 :
All P is either C or D or E ;
S is neither C nor D nor E ;
therefore, S is not P.
xxi.] QUESTIONS AND ANSWERS. 197
The premises are respectively :
P - PC -I- PD -I- PE.
S = Scde.
We get the conclusion in the briefest way by multiplying
the two premises together as they stand ; thus :
PS = P (C ! D -I- E) Scde = o -|- o .|. o.
Each alternative is found to be contradictory, so that
there is no such thing as PS, that is to say, no P is S.
The argument is not, however, correctly described by
Dr. Thomson as in the syllogistic mood U E E, nor are the
other forms of argument given in the same section syllo
gistic. They are disjunctive in character.
24. If Abraham were justified, it must have been
either by faith or by works : now he was not
justified by faith (according to James), nor by
works (according to Paul) : therefore Abraham
was not justified. [w.]
There is some difficulty in deciding on the best method
of symbolising this argument, owing to the vagueness of
the conditions when analysed ; but the following seems to
be the best representation :
Let A = Abraham ; C = justified by works ;
B = justified ; D = justified by faith.
Then the premises are :
AB - AB (C -I- D).
A = Ac.
A = Ad.
198 EQUATIONAL LOGIC. [CHAP.
These premises will be found to erase all the combina
tions of A excepting hbcd, which gives the conclusion.
The combinations of a are altogether unaffected and need
not be examined. The student may try other modes of
representing the premises, but should get A = Kb by every
method.
25. It must be admitted, indeed, that (i) a man
who has been accustomed to enjoy liberty
cannot be happy in the condition of a slave :
(3) many of the negroes, however, may be
happy in the condition of slaves, because
(2) they have never been accustomed to enjoy
liberty. [w.]
Let A = man accustomed to enjoy liberty ;
B = happy in condition of slave ;
C = certain negroes.
The premises may be stated in the forms
A - Kb. (i)
C = 0C. (2)
The supposed conclusion is C = CB. (3)
The possible combinations are as in the margin, from
Me
abC
abc
which it will be seen that
C = tfBC ! abC;
that is, are either B, happy, or b, not
happy.
The fallacy is that of Negative Premises or of Illicit
Process of the Major.
XXL] QUESTIONS AND ANSWERS. 199
26. If that which is devoid of heat and devoid
of visible motion is devoid of energy, it
follows that what is devoid of visible motion
but possesses energy cannot be devoid of
heat.
Let A = possessing heat ;
B = possessing visible motion ;
C = possessing energy,
the universe being things unexpressed, and devoid of
being taken as the negative of possessing. Then the
condition is :
ab = abc.
By contraposition we obtain, using Mr. MacColl s notation
for the negative of ab (See Preface) :
C = C (ab)
= C (A3 -I- a~B -I- AB)
Hence bC =
two self -contradictory alternatives disappearing. It can
also be readily shown that this inference is equivalent to
the original condition.
27. Prove the logical equivalence of the pro
position B .= AC ] ac and b = Ac [ aC.
This might be shown by receding to the combinations of
the Logical Alphabet, but it is more neatly proved by
equating the negatives of each member of the first equation.
If M = N, then also m n (p. 184); hence the negative
of B must be identical with the negative of AC ! ac.
Now the negative of B is b ; that of the compound and
200 EQUATIONAL LOGIC. [CHAP.
complex member is the compound of the negatives of the
two alternatives. In Mr. MacColl s notation
(AC -I- ac) = (AC) (ac)
- (flC "I" Ac ! ac) (Ac -I- *C I AC).
On multiplying out, the nine products are found to be all
self-contradictory excepting 0C I* Ac, which is therefore
the expression for b. Vice versa the negative of Ac -\- aC
will be found to be AC -|- ac, so that the propositions are
clearly equivalent.
28. If no A is BC, what can I infer about the
relation of B and AC ?
The condition is
A = Ab -I- Ac.
Substitute in either side of
ABC = ABC,
and we get
ABC - ABZ>C -I- ABG: - o,
or B cannot be AC.
29. It is known of certain things that (i) where
the quality A is, B is not ; (2) where B is, and
only where B is, C and D are. Derive from
these conditions a description of the class of
things in which A is not present, but C is.
The premises are clearly :
(1) . . A = Ab;
(2) . . B = CD
xxi.] QUESTIONS AND ANSWERS. 201
The conceivable constituents of the class which is C
but not-A are
aC = tfBCD -I- aECd -|- abCD >\- abCd.
Substituting CD for B in the second, and B for CD in
the third alternative, we find that these combinations give
contradictory results, namely aBCDd and 0BCD. It
follows that
aC = 0BCD -I- abCd.
Observe that the premise (i) has no connection with this
result, which is deducible from (2) alone.
30. It has been observed that in a certain class
of substances, (i) where the properties A and
B are present, the property C is present ; and
(2) where B and C are present, A is present.
Does it follow that B is present where C and
A are present ?
The premises are obviously :
(1) . . AB = ABC,
(2) . . BC - ABC,
which are equivalent to the single proposition
AB - BC.
The answer is obtained at once from I ABC
the combinations in the margin, which
show that
AC = ABC -I- KbC ;
that is, in the presence of A and C, B
is indifferently present or absent. [Dr.
Macfarlane (Algebra of Logic, 1879, p. 141), who gives this
KbC
Ate
abC
abc
202 EQUATIONAL LOGIC. [CHAP.
problem, requires more than half a page to solve it, using,
moreover, sundry impossible logical fractions.]
31. Given (i) that everything is either B or C,
and (2) that all C is B, unless it is not A :
prove that all A is B. [c.]
The first condition is expressed by the assertion that
every not-B is C, which carries with it the equivalent that
every not-C is B. Thus we have
b = bC. (i)
The second assertion is less easy to interpret, because we
are not told what happens if it, that is C, is not-A. The
meaning appears, however, to be that C if it is A must
be B, that is
AC = ABC. (2)
These conditions give the combinations
ABC.
abC.
from the first two of which we learn that A is always B.
32. If we throw every A is B into the form
every A is B or B/ we have every A which is
not B is B a contradiction in terms. But it
evidently implies that there can be no As which
are not Bs, and thus we return to every A
is B.
xxi.] QUESTIONS AND ANSWERS. 203
The above is a transcript with altered symbols from
De Morgan s sixth example (Formal Logic, p. 123). But
the contradiction arises simply from an error in not multi
plying both alternatives by b. De Morgan follows the rule
for the resolution of dilemmas, not observing that this
rule can apply only when the alternatives are different.
Equationally we have
A = AB -I- AB.
hence . . . Kb = AB -|- AB = o
he gets . . . A3 = AB -|- AB - AB.
It is rarely we find De Morgan tripping.
33. Every A is one only of the two B or
C ; D is both B and C, except when B is
E, and then it is neither ; therefore no A
is D.
This problem was proposed by De Morgan in the Formal
Logic, p. 124, and a solution has been given in the Principles
of Science, p. 101 (first ed., vol. i. p. 117). The premises,
as Professor Groom Robertson has pointed out, may be
stated in two propositions, namely
A = AEc -I- AC.
D = D<?BC -I- DEfo
Some objection has, however, been taken by Mr. Monro
to my solution, and the student will find a good exercise
in going over the solution carefully. It seems rather doubt
ful how we should treat the combinations which are E
and not B ; but the difficulties lie wholly in the interpreta
tion of De Morgan s conditions.
204 EQUATIONAL LOGIC. [CHAP.
34. From A follows B, and from C follows D ;
but B and D are inconsistent with each other.
Hence A and C are inconsistent with each
other.
This problem which is formally the same as one of
De Morgan s (Formal Logic, p. 123, Example 3), has the
conditions
(1) A = AB;
(2) C-CD;
(3) B = B*/.
The consistent combinations are
abCD.
abcD.
abed.
We see that A and C never occur together, and in fact
that A is never found excepting in the presence of B and
the absence of both C and D.
I committed an error in treating this problem in the
Substitution of Similars (pp. 52, 53), by regarding a&cd
as negatived by the premises. B may occur in the absence
of A, but C and D must both be absent.
35. What are the combinations of the qualities
A, B and C which are possible according to
the following conditions? (i) Where A is
present, B and C are either both present at
once or absent at once ; (2) where C is present,
A is present. Describe the class not-B under
these conditions.
QUESTIONS AND ANSWERS. 205
The conditions are expressed equationally as
(1) A = ABC -I- Mf.
(2) C = AC.
The consistent combinations are shown in the margin ;
ABC
abc
and A^C are removed by the first condition,
and #BC and abC by the second. Selecting the
two remaining ones which contain , we have the
required discription
b = hbc { abc = be.
Where B is absent, C also will be absent.
36. The logical value of two affirmative pre
mises in the second figure is absolute zero.
Examine the truth of this statement. [P.]
The two premises assumed to be universal may be
symbolised as
(1) A = AB;
(2) C - CB.
The first negatives the combinations AC and Kbc, the
second AC and abC, so that the premises overlap in
regard to A^C. There remain five combinations. If we
inquire what is A we get the value
A = ABC ] KEc = AB,
which is no more than (i ). For the description of C similarly
we get (2). Thus it is plain that no relation is established
between A and C. Concerning B we have even less
information; for
B = ABC -I- AB^ -I- rtBC -|- aEc = AB -|- aB = B.
206 EQUATIONAL LOGIC. [CHAP.
Of the negative terms, however, we draw more significant
descriptions ; thus
a = aEC -I- ac.
b abc.
c = EC -|- abc.
It cannot be truly said that the logical value of the
premises is absolute zero.
37. Given that (i) everything which is B but not
D is either both A and C or neither A nor C ;
and (2) that neither C nor D is both A and B:
prove that no A is B.
[Adapted from Moral Science Tripos,
Cambridge, 1879.]
The conditions are
(1) B^=B^(AC -I- ac).
(2) C -I- D = (C ! D) (a -I- b).
Confining our attention to the combinations containing
AB, we see that ABcd is contradicted by (i), and the
rest which contain either C or D, by (2). Hence there are
no ABs, or no A is B.
The equation (2) may be more briefly stated as
AB = AB*/.
The only combination containing a removed by (i) and
(2) is 0BG
38. Illustrate the use of symbolic methods by
expressing the propositions
xxi.] QUESTIONS AND ANSWERS. 207
(1) No A is B except what is both C and
D, and only some of that.
(2) Either C or D is never absent except
where A or B is present, but both are
always absent then. [C.]
The first proposition appears to deny the presence of any
combination containing AB except there be also present C
and D, and only in some cases then. To express this some
we must introduce another letter term, say E, so that where
E is present the above holds true ; where E is absent, A is
not B at all. We find then that the following combinations
are negatived :
ABG/E. ABCD^.
ABdDE. ABGfc.
ABa/E. AEcDe.
All this may be expressed in the one equation
AB = ABCDE.
The proposition (2) is not easy to interpret, but seems to
mean
A -I- B = cd.
39. Every X is either P, Q, or R ; but every P
is M, every Q is M, every R is M ; therefore
every X is M.
De Morgan, who gives the above (Formal Logic, p. 123,
2 o8 EQUATIONAL LOGIC. [CHAP.
Example 5), describes it as a common form of the dilemma.
It is thus solved equationally :
(1) X = X(P -I- Q -I- R);
(2) P = PM;
(3) Q= QM;
(4) R = RM.
Substituting by (2) (3) (4) in (i),
X = X(PM -I- QM -I- RM);
X = X (P -I- Q -I- R) M.
Re-substituting in the last by (i)
X - XM.
40. Every A is either B, C, or D ; no B is A ;
no C is A ; therefore, every A is D.
[De Morgan, Formal Logic, p. 122.]
The premises are clearly
(1) A = AB -I- AC -I- AD.
(2) B = 0B.
(3) C = C.
In (i) substitute the values of B and C given in (2) and
(3), and then strike out two self-contradictory terms
A == AflB ! A0C ! AD - AD.
41. If A be B, E is F ; and if C be D, E is F ;
but either A is B, or C is D ; therefore, E is
F. (De Morgan, Formal Logic, p. 123.)
xxi.] QUESTIONS AND ANSWERS. 209
This appears to be more complicated in symbols than it
really is. The first two premises are
(1) AB = ABEF
(2) CD - CDEF.
To express the third premise we must introduce explicitly
the tacit term, say X, meaning the circumstances under
which the proposition holds good, in this place, or at
this time, or under certain assumed conditions. Thus we
have
(3) X - XAB -I- XCD.
substituting by means of (i) and (2),
X - (XAB -I- XCD)EF,
and re-substituting by (3)
X = XEF.
42. Every A is either B or C, and every C is A.
This, says De Morgan (ibid. p. 123), is wholly
inconclusive, and leads to an identical result.
Equationally treated this is not quite so. The premises
are
(i) A - AB -I- AC
(a) C = AC
Hence (3) A - AB -|- C.
De Morgan finds that Kb is C, which C being A gives
A is A a necessary proposition or truism. But we also
get, multiplying each side of (3) by b,
Kb = AB -I- bC = bC.
In the absence, then, of B, there is identity between A
and C, but in the presence of B, A may be either B or C.
P
210 EQUATIONAL LOGIC. [CHAP.
43. Every A is B or C or D ; every B is E ; every
C is E ; and every E is D.
[De Morgan, ibid. p. 123, Example 4.]
Thus symbolised
(i) A - AB -I- AC -I- AD.
(2) B = BE. (3) C = CE. (4) E = ED.
By obvious substitutions, by (2) and (3) in (i), and then
by (4) in the result, we get
A ="ABDE -I ACDE ! AD.
But the first two of these alternatives are superfluous ;
they both involve D and are therefore contained in the wider
term AD. Hence
A = AD.
44. * If the relations A and B combine into C, it
is clear that A without C following means that
there is not B, and that B without C following
means that there is not A.
[De Morgan, Third Memoir on the Syllogism, p. 48.]
The relations A and B combining into C appears to mean
simply that AB is accompanied by C, or
AB = ABC.
To find A without C following, we have necessarily
Kc - ABr -I- Afo
Inserting for AB in this last its value ABC.
Kc = ABG: i Kbc = Kbc.
Similarly for B without C following
B<: = AB<: -I- d&c = ABG: I c&c = d&c.
xxi.] QUESTIONS AND ANSWERS. 211
45. Suppose a class S to be divided (i-) on one
principle into A and B, and on another prin
ciple (2) into C and D, the divisions being
exhaustive ; suppose further that (3) all A is
C, and (4) all B is D ; can you conclude that
all C is A, and all D is B ? [E.]
The meaning of this problem appears to be that the class
S will, as regards A and B, consist of SA$ and S#B, and
similarly as regards C and D ; if so, there will under the
first two conditions be only four possible combinations,
namely
SAJCtf. SaBG/.
SAM). SaBd).
But the further condition (3) negatives SA&rD, and (4)
negatives S#BC</, so that, on inquiring for the description
of C, we find it is (within the class S), A.bCd- } similarly D
is a~BcD, Both questions then may be answered in the
affirmative, provided that we are not to look beyond the
sphere of the class S.
46. What are the classes of objects regarded as
possessing or not possessing the qualities
A, B, C, D, which may exist consistently
with the fundamental Laws of Thought, and
the conditions that no class possesses both
A and B, and that everything which does not
possess B possesses C but not D ? [L.]
The first condition that no class possesses both A and B
will be sufficiently expressed in the premise A = A, which
212
EQUATIONAL LOGIC.
[CHAP.
prevents A and B from meeting. The second condition is
obviously b = bCd. On going over the sixteen combina
tions in the fifth column of the Logical Alphabet (p. 181),
it will be obvious that the first four, containing AB, are
negatived by the first premise. The third four
(#B) remain untouched : of the second and fourth
f ours con taining , all are negatived except KbCd
an( * a ^d> The adjoining list of combinations is
abCd therefore the answer to the question.
h.bCd
BCD
aBCd
AB
47. How can we represent analytically the precise
meaning of the opposition between a universal
affirmative proposition and its contradictory,
say between All As are Bs, and some As are
not Bs ?
The universal affirmative is symbolised as A = AB, and
its logical power is to negative the combination A, as
shown in the margin. Now some As are Bs
was before explained to mean one A at least, it
may be more or all As. But, even if there be
one Ab found, it establishes the existence of the
combination, subject to remarks elsewhere made
(p. 142). In this qualitative treatment of logic number
enters not at all, so that one counts for as much as a
million. The force of the particular negative proposition is,
then, to restore the combination which had been removed
by the universal affirmative.
48. If to the premises of an affirmative sorites we
add a proposition affirming the first subject of
ab
xxi.] QUESTIONS AND ANSWERS. 213
the last predicate, the conditions now become
equivalent to an equally numerous series of
identities, or doubly universal propositions in
Thomsons 5 form U.
Symbolically, if we have the series of premises A = AB,
B = BC, C = CD, and so on, up to X = XY, and we then
add the condition Y = AY, the premises immediately
become the same in logical force as
A = B = C = D=.... = X = Y.
To give a perfect demonstration of this theorem might
not be very easy ; but the student may convince himself of
its truth by observing in several trials that the combinations
consistent with the premises of a sorites as shown above,
never contain a negative letter to the right hand of a positive
one in the usual order of the alphabet. Thus the com
binations consistent with the first two premises are
ABC, aBCy abC and abc\
those for the first three are
ABCD, rtBCD, 00CD, abcD and abed.
Hence the last predicate appears in every combination
except the last, and the first subject only in the first
combination. In affirming the first subject of the last
predicate, then, all combinations except the first, which
contains both terms, and the last, which contains neither,
must disappear. There remain in every case only the two
combinations ABCDE .... XY .... and abcde ....
xy . . . . which proceed from the identities stated in the
question.
2i 4 EQUATIONAL LOGIC. [CHAP.
Suppose a pillar of circular section to be so shaped that
no lower section is of less diameter than any upper section,
but the section at the bottom is not greater than the section
at the top ; we have here a physical analogue to the heap
of propositions described above.
49. Is Professor Alexander Bain correct in the
following extract from his Deductive Logic
(P- 159)?
( T ) Socrates was the master of Plato.
(2) Socrates fought at Delium.
(3) The master of Plato fought at Delium.
* It may fairly be doubted ^whether the transitions, in this
instance, are anything more than equivalent forms. For the
proposition (4) " Socrates was the master of Plato, and
fought at Delium," compounded out of the two premises, is
obviously nothing more than a grammatical abbreviation.
No one can say that there is here any change of meaning,
or anything beyond a verbal modification of the original
form.
Professor Bain in writing the above was clearly in need
of means of more accurate analysis than his logical studies
had afforded him. For if we put
A = Socrates ; B = master of Plato ;
C = one who fought at Delium,
the premises are certainly
(1) A = B;
(2) A = AC.
The conclusion (3) as it stands is B = BC, which negatives
only two combinations AB<: and aBc, whereas the premises
xxi.] QUESTIONS AND ANSWERS. 215
negative in addition the three A^C, Afc, aRC. It is possible,
indeed, to draw the conclusion (5)6 = AC, which is better
than (3) by two combinations, namely, A&C and BC. As
to the supposed proposition (4), it cannot be made into any
non-disjunctive proposition without a change of meaning ;
for, whether we make it into (6) A = BC, or A = ABC, it
differs in force from the premises (i) and (2), which propo
sitions in fact cannot be condensed into any single non-disjunc
tive proposition of equivalent meaning. The fact is that the
supposed proposition (4) consists merely of the two (i) and
(2) re-stated in one compound sentence. It is not the
proposition at all; it is the propositions. The case
would have been considerably altered, indeed, had Mr.
Bain interpreted (i) as Socrates was a master of Plato, of
the form A = AB. The type of the premises would then
have been essentially altered; but that he does not so inter
pret it is obvious from (3), in which we have the master,
not a master. Altogether this page of Professor Bain s
work affords remarkable evidence of the inability of a most
acute logician to maintain accuracy of logical vision without
the aid of some kind of calculus like that developed in the
latter part of this work.
I append logical diagrams which almost explain them
selves, the combinations pointed out by each bracket being
those negatived by the proposition whose number is attached
to the bracket.
ABC
(2) : ,
"U-
(3)
abC
abc
216 EQUATIONAL LOGIC. [CHAP.
50. How far does the conclusion of an Aristo
telian syllogism fall short of giving all the
information contained in the premises ?
The premises of Barbara, say A = AB, B = BC, nega
tive four combinations, AB<r, A$C, A&*, a&c. The conclusion
A = AC negatives only two of these, namely, ABc and
AJbc. Measured in this way, then, it contains only half of
the information of the premises; but of course if the
conclusion gives just that information which is desired, the
overlooking of the rest is no harm. Enough is as good as
a feast or rather better.
51. Take the premises of a syllogism in Barbara,
such as (i) all As are Bs, and (2) all Bs are
Cs, and determine precisely what they affirm,
what they deny, and what they leave in
doubt.
ABC
\~Kbc
aQC
To answer this question, we must form
the eight combinations of A, B, C and
their negatives, as in the margin; we
then strike out AC and hbc as being in
conflict with condition (i), and AB<r and
a~Bc as being similarly in conflict with the
condition (2), that all Bs are Cs. There
remain four combinations, ABC, #BC, abC, and abc. But
these do not stand on the same logical footing, because if
we were to remove ABC, there would be no such thing as
A left ; and if we were to remove abc there would be no
such thing as c left. Now it is the Criterion or condition of
logical consistency (p. 181) that every separate term and its
abC
abc
xxi.] QUESTIONS AND ANSWERS. 217
negative shall remain. Hence there must exist some things
which are described by ABC, and other things described by
abc. But as regards the remaining two combinations, #BC
and abC, the case is different; for we may remove either, or
both of these without wholly removing any term. We
might add to the premises the new condition that all BCs
are As, or BC = ABC, which would negative #BC ; or we
might add the condition, all Cs are As, or C = AC, which
would remove both 0BC and abC.
We may sum up the meaning of the original premises (i)
and (2) by saying that they deny the existence of AB<r,
AC, hbc, and aEc ; that they affirm the presence or logical
existence of ABC and abc; and thirdly, while leaving #BC
and abC uncontradicted, they are consistent with the presence
or absence of these two combinations. This is all that they
leave in doubt concerning the relations of A, B, and C.
52. What is the amount of contradiction in the
following celebrated epigram ?
The Germans in Greek,
Are sadly to seek ;
* * * *
All, save only Hermann,
And Hermann s a German.
Putting A = German ; B = Hermann ;
C = sadly to seek in Greek,
the premises are evidently
(1) A = AC.
(2) B = B*
(3) -B = AB.
218 EQUATIONAL LOGIC. [CHAP.
The logical diagram is as in the margin; it will be
noticed that B disappears entirely, indi-
eating contradiction ; but A remains in
the combination AC, It is obvious that
the wit of the epigram arises from the per
ception of contradiction. (See Hamilton s
Lectures , vol. iii. p. 393.)
ABC
BC
(2)
abC
abc
53. Show that you can make no assertion about
two terms A and B (and these only), which
is not either contained in the assertion of
identity (A = B), or else contradictory thereto.
The proposition A = B removes two out of the four
combinations thus
Consistent Inconsistent
Combinations. Combinations.
AB. A.
ab. tfB.
Now, if any new assertion negatives either or both of Kb
and flB, it must be an assertion contained in and inferrible
from A = B. If it removes either AB or ab, it must con
tradict A = B, because either A and B or a and b will then
disappear entirely from the Logical Alphabet. It might be
said perhaps that a new assertion could remove one consis
tent and one inconsistent combination, for instance, ab and
Ad; but this cannot be done except by a contradictory
assertion. Any other pairs such as AB and Kb, AB and
0B, or ab and a~B, being removed, removes some letter
entirely and involves contradiction.
xxi.] QUESTIONS AND ANSWERS. 219
54. Is it (i) logically (2) physically possible that
all material things are subject to the law of
gravity, and that at the same time all not
material things should be subject to the same
law ? [L.]
It is logically possible, that is to say, in accordance with
the Laws of Thought, that all things material and all things
not material should be subject to the law of gravity. In
this case what is not subject to the law of gravity would be
found among not-things. But it is not logically possible
that all (material things) and all not-(material things) should
be subject to the law of gravity, because this is equivalent
to denying the existence of any class not subject to the law
of gravity. This class would by one condition be not-
material, and by the other condition it would be material,
which is impossible. But by the law already described
(p. 181) as the Law of Infinity, every logical term must
be assumed to have its negative. The student is recom
mended to work out this question with the aid of letter
symbols.
As to the second part of the question, what is not logically
possible is of course not physically possible. Hence we are
restricted to the inquiry whether it is physically possible that
all things material and all things not-material should be
subject to the law of gravity. This can only be answered
on logical grounds thus far, that if the property of gravita
tion is essential to material things and forms a part of the
definition of them, then it is not possible that not-material
things should gravitate. As a matter of fact the possession
of inertia is perhaps the ultimate test of materiality ; but
gravity is proportional to inertia and is an equally good
test.
220 EQUATIONAL LOGIC. [CHAP.
55. It is observed that the phenomena A, B, C
occur only in the combinations ABc, abC, and
dbc. What propositions will express the laws
of relation between these phenomena ?
Of the eight combinations of A, B, C, only these three
remain. As we see that A occurs with and only with B, and
a with and only with #, it is firstly obvious that A = B is
the chief law. But as this law of relation leaves the com
bination ABC uncontradicted, we must have a second law
to remove this, which may be either AB = AB<r, or else
B = B<r. Observe, however, that the laws A = B and
B = B<: overlap and are pleonastic, because they both
deny that B can be #BC. Hence the simplest statement
of the laws of relation is
A - B.
AB - ABr.
56. Given three terms, for instance, water, blue,
and fluid, how would you proceed to ascertain
the utmost number of purely logical relations
which can exist among them ? [L.]
The relations of any three terms or things or classes of
things must be governed in the first place by the universal
Laws of Thought (p. 180). These laws restrict the combi
nations of three things, present or absent, to eight at the
utmost; for each thing may be present or absent giving
2x2x2 = 8 cases. But any special logical relation
which may exist between the things has the effect of
further restricting these combinations ; the relation that
water is a fluid, prevents the existence of the combination
xxi.] QUESTIONS AND ANSWERS. 221
water, not-fluid. Conversely the removal from the series of
any one or more of the eight combinations expresses the
existence of a relation or relations negativing the existence
of these combinations. Thus, the removal of the two
combinations water, not-blue, fluid ; water, not-blue, not-
fluid, expresses the law that all water is blue. Thus the
logical meaning of any condition is represented by the state
of the combinations agreeing with those conditions. It
follows that the utmost possible number of distinct logical
relations will be ascertained by taking the eight possible
combinations of the three terms and striking out one or
more of the combinations in every possible variety of ways.
The number of these ways cannot exceed 256 ; for each of
the eight combinations may be either present or absent,
giving 2x2x2 x 2 X2 x 2 x 2x 2 = 256 ways.
But this calculation will include many cases where one or
more of the three terms and their negatives disappear alto
gether, representing contradiction in the conditions. Many
different selections, too, proceed from logical relations
similar in character and form ; thus the law A = AB is
similar to A = A, and to a ab ; the law A = BC -|- be
is similar to C = AB ) ab ; and so forth. The investiga
tion is fully described in the Principles of Science (pp. 134-
143 j ist ed. vol. i. pp. 154-164) as also in the Memoirs of
the Manchester Literary and Philosophical Society^ Third
Series, vol. v. pp. 119-130. It is found that the 256
possible selections are thus accounted for
Proceeding from consistent logical conditions 192
inconsistent 63
no condition at all i
256
222
EQUATIONAL LOGIC.
[CHAP. xxi.
The consistent logical conditions are found, however, on
careful analysis to fall into no more than fifteen distinct
forms, or types of relation, which are stated in the following
table
Reference
Number.
Propositions expressing the
general type of the logical
conditions.
Number of dis
tinct logical
variations.
Number of
combinations
contradicted
by each.
I.
A = B
6
4
II.
A = AB
12
2
III.
A = B, B = C
4
6
IV.
A = B, B = BC
24
5
V.
A = AB, B = BC
24
4
VI. A = BC"
2 4
4
VII. j A = ABC
2 4
3
VIII. ! AB = ABC
8
i
IX. ! A - AB, B = aEc
2 4
3
X. A = ABC, at = abC
8
4
XL
AB = ABC, al> = abc
4
2
XII. AB = AC
12
2
XIII.
A = BC ! Kbc
8
3
XIV.
A - BC ! be
2
4
XV.
A = ABC, a = aEc -\- abC
8
5
i
CHAPTER XXII.
ON THE RELATIONS OF PROPOSITIONS INVOLVING THREE OR
MORE TERMS.
i. THE doctrine of the opposition of propositions, exhi
bited in the well-known square, is an important and inter
esting fragment of ancient logic; but it is now apparent
that propositions involving only two terms, one subject and
one predicate, do not sufficiently open up the question of
the relationship of propositions. Two terms admit of only
four combinations, and these can be present and absent
only in sixteen ways, nine of which involve contradiction.
There remain only seven cases of logical relation which
resolve themselves into only two distinct types of propo
sition. (Principles of Science, pp. 134-7; ist ed. vol. i.
p. 154-7.) With the introduction of a third term the
sphere of inquiry becomes immensely extended. There
are now, as we have seen (p. 221) 193 different cases of
selection of combinations resolving themselves into fifteen
distinct types of relation. The possible modes of relation
of one proposition to another, including under the expres
sion one proposition any group of propositions, become
considerably complex. Such modes of relation seem to
be seven in number : thus one proposition is as regards
another
224 EQUATIONAL LOGIC. [CHAP.
(1) Equivalent.
(2) Inferrible, or contained in the other, but not equi
valent.
(3) Partially inferrible and otherwise consistent.
(4) Consistent but indifferent and not inferrible.
(5) Partially inferrible, partially contradictory.
(6) Partially indifferent, partially contradictory.
(7) Contradictory.
2. Let us take as an example the proposition
Steam = aqueous vapour,
and give a pretty complete analysis of its related pro
positions.
Let A = steam ;
B = aqueous ;
C = vapour.
The proposition being evidently of the form
(i) A = BC,
the combinations contradicted will be as in the margin.
ABC The equivalent proposition will be
Not steam = not aqueous or not vapour.
An inferrible but not equivalent assertion
will be any one which negatives one, two,
or three, but not four of the combinations
negatived by (i). There will therefore be
4x3 4x^x2 * *
4 + - - + or 1 4 such infer-
1x2 1x2x3
rible and logically distinct propositions. We may infer
steam is aqueous ; steam is vapour ; what is not vapour
is not steam; what is not aqueous is not steam; non-
aqueous vapour is not steam ; and so forth.
XXIL] RELATIONS OF PROPOSITIONS. 225
The third class of related propositions will include those
which negative one or more of the excluded combinations,
and one or more indifferent combinations. Indifferent
combinations, as the name expresses, are those which can
be removed without wholly removing any of the letters
A, B, C, a, b, c. In this case any one of the remaining
combinations except ABC may be singly removed. Thus
not steam is not aqueous, or in letters a a b, is not
contradictory to (i) and it may be inferred from (i) in
respect of vapour which is not steam. But the assertion
that other things which are not steam are not aqueous is
not inferrible, but is consistent with (i). A proposition,
again, which should negative A^C, Kbc, aRc, abC will be
inferrible in respect of the two former, and consistent in
respect of the two latter combinations. To ascertain what
such proposition is we must look in the Logical Index,
afterwards described, for the proposition which leaves a fi e 0,
and we find in the 55th place b ac, or not-aqueous =
not-steam and not-vapour. The other possible propositions
of the same class are numerous and various.
To obtain one of the fourth class, which is merely con
sistent and indifferent, we must take any one or more of
the combinations unnegatived by (i), for instance a&C, in
such a way as not wholly to remove any letter. Thus
ab = abc, or not-steam which is not aqueous is not vapour
is an assertion quite indifferent to (i). So is the assertion
#B = #C (Logical Index, No. 7).
Contradictory propositions being defined as those which
wholly remove any term, such will be any one which re
moves ABC. Thus to say that steam is not aqueous is a
case of the 5th class; it is inferrible from (r) in respect of
steam which is not vapour (AB^), but it is contradictory
because it also negatives steam which is vapour.
Q
226 EQUATIONAL LOGIC. [CHAP. xxn.
A proposition of the sixth class is discovered by taking
any combination which may be spared with one which
cannot, such as abC and ABC, and looking in the index,
we find AC - ^C, or steam- vapour is identical with non-
aqueous vapour, as a partially consistent, partially contra
dictory proposition as regards (i). It may or may not be
true that what is not steam and not aqueous is not vapour,
but it is contradictory to (i) to say that vaporous steam is
not aqueous.
An example of a simply contradictory proposition of
Class 7 is found in one which removes ABC only, such as
AB = ABr ; again a = aE, or not-steam is aqueous deletes
b ; c = AEc deletes c.
2. As a second example, let us take the propositions
(i) Hand = right-hand or left-hand; (2) Right is not left.
Putting A = hand; B = right; C = left; the conditions
are evidently
(1) A = AB -I- AC.
(2) B = EC.
The consistent combinations are shown in the margin,
ABC and the student ma y verify the following
list, which gives one specimen of each of
(2) the seven classes of related assertions, the
reference number of the Logical Index
being also added.
(1) Equivalent. B = EC; be = abc. No. 153.
(2) Inferrible, etc. aE = aEc. No. 9.
(3) Partially inferrible, etc. a = abc. No. 15.
(4) Consistent, etc. AB - ABC; ab = abc. No. 67.
(5) Partially inferrible, etc. C = AC; A = AB. No. 59.
(6) Partially indifferent, etc. A - AB^; ab = abc. No. 179.
(7) Contradictory, b = be. No. 35.
dnC
abC
abc
CHAPTER XXIII.
EXERCISES IN EQUATIONAL LOGIC.
I NOW give a small collection of examples and problems
designed to enable the student to acquire a complete com
mand of the equational and combinational views of logic.
They are for the most part devised specially for this book,
but a few have been utilised in examination papers, and a
few have been adopted as indicated from the papers of other
examiners. These questions form perhaps a partial answer
to Professor Sylvester s remark, as quoted in the preface,
especially when we observe that the questions and problems
involving the relations of three terms can be multiplied
almost ad tnfinttum, without resorting to like questions in
volving four, five, or more terms. The student will readily
gather that the number, variety, and complexity of problems
in pure logic is simply infinite, and is such as we gain no
glimpse, of in the old Aristotelian text-books.
i. Represent equationally the following assertions :
(1) With the exception of porcelain there is no non-
metallic substance which has been employed to
make coins.
(2) With the exception of zinc and the metals discovered
during the last hundred years, there is no metal
which has not been employed to make coins.
228 EQUATIONAL LOGIC. [CHAP.
(3) The worth of that is that which it contains,
And that is this, and this with thee remains.
[SHAKESPEARE.]
(4) It is dangerous to let a man know how far he is but
a brute, without showing him also his grandeur.
It is dangerous again to let him see his grandeur,
without his baseness. It is [even more] dangerous
to leave him ignorant in both ways ; but it is a
high advantage to represent to him both the one
and the other. (Pascal, Pensees.)
2. Represent in the forms of equational logic any of the
following arguments :
(1) Milton was a great poet, and a fearless opponent of
injustice ; a great poet should be honoured ; a
fearless opponent of injustice should be honoured :
therefore Milton should be honoured.
(2) The virtues are either passions, faculties, or habits:
they are not passions, for passions do not depend
on previous determination ; nor are they faculties,
for we possess faculties by nature ; therefore they
are habits.
(3) There can be no person really fit to exercise absolute
power, because the qualifications requisite to fit a
person for such a position would consist in native
talent combined with early training ; now such a
talent cannot be possessed in early childhood.
(Suggested by De Morgan, Syllabus, p. 67.)
(4) One of the masters of chemistry was Berzelius ;
Berzelius was a Swede ;
One of the masters of chemistry was a Swede. [D.]
(5) This heavenly body is either a planet or a fixed star ;
xxiii.] EXERCISES. 229
all fixed stars twinkle; planets do not twinkle;
this body twinkles, therefore it is a fixed star.
(6) Show me any number of men, and I will say with
confidence, either that they will with one accord
raise their voices for liberty, or that there are
aliens among them. (The stump orator s mode,
according to De Morgan, of saying that all
Englishmen are lovers of liberty.) [B.]
3. Infer all that you possibly can, by way of contra
position or otherwise, from the assertion all A that is
neither B nor C is D. [R.]
4. Express equationally Miscellaneous Example No. 39
in Elementary Lessons in Logic ; p. 317.
5. What proposition concerning nebulas and vaporous
bodies leaves doubtful the existence of a class of things
which are neither nebulae nor vaporous bodies ?
6. Represent the fact that A differs from B in two equiva
lent equational propositions.
7. Prove equationally that the proposition, All elements
are either metal-elements or elements, is a mere truism.
8. What is the difference between the propositions
A = AB -|. A, B = AB .|. B, and A = B .|. A?
9. Prove that if all not-Bs are not-As, and all Bs are As,
then A = B, and vice versti.
10. Show that the negative premises No As are Bs and
no Cs are Bs, imply the logical existence of a class B which
is neither A nor C.
1 1. Prove the equivalence of the following assertions :
(1) Every gem is either rich or rare.
(2) Every gem which is not rich is rare.
(3) Every gem which is not rare is rich.
(4) Everything which is neither rich nor rare is not a gem.
230 EOUATIONAL LOGIC. [CHAP.
12. Show that if metals which are either not valuable or
not destructible are unfitted for use as money, it follows
that destructible metals which are fitted for use as money
must be valuable.
13. Does the proposition A = B -|- BC differ in force
from A = B ?
1 4. All animals having red blood corpuscles are identical
with those having a brain in connection with a spinal cord.
What is the description you may draw from this proposition
of things having a brain not in connection with a spinal
cord ?
15. Luminous body is either self-luminous or luminous
by reflection ; melted gold is both self-luminous and lumi
nous by reflection. Unmelted gold is not self-luminous
but is luminous by reflection. Represent these premises
symbolically, and draw descriptions of the terms (i) lumi
nous body, (2) self-luminous body, (3) body luminous by
reflection, (4) body not luminous, (5) body not self-luminous,
(6) not melted gold, (7) not unmelted gold.
1 6. There are no organic beings which are devoid of
carbon. Determine precisely what this proposition affirms,
what it denies, and what it leaves doubtful.
17. Prove the equivalence of the following statements
No right-angled triangles are equilateral; no equilateral
triangles are right-angled ; no right-angled equilateral figures
are triangles.
1 8. All scalene triangles have their three angles equal to
two right angles. What are the least or simplest assertions
which added to the above will make it equivalent to All
triangles are all figures which have their three angles equal
to two right angles ?
xxiii.] EXERCISES. 231
19. All equal -sided squares have four right angles.
What is the least extensive proposition which added to the
above makes it equivalent to All squares are equal-sided
and have four right angles ?
20. If an orator were to assert that Afghanistan is a very
poor country, but it is essential to the security of India, but
a reporter were to consolidate these two assertions into the
one assertion that a very poor country, Afghanistan, is the
Afghanistan which is essential to the security of India, how
far would the reporter have misrepresented the logical
meaning of the orator ?
21. Express the following argument equationally : Every
organ of sense has nervous communication with the brain ;
for such is the case with all the five organs of sense, the
eye, ear, nose, tongue, and skin.
22. If requested to draw from the assertion All coal
contains carbon a description of the term metal, what
answer would you give ?
23. What values will you obtain for the terms man, brute,
and gorilla, under the conditions that a gorilla is a man, and
that all men are included and all gorillas excluded from the
class of non-brutes ?
22. Assuming that armed steam-vessels consist of the
armed vessels of the Mediterranean and the steam-vessels
not of the Mediterranean, inquire whether we can thence
infer the following results :
(1) There are no armed vessels except steam- vessels in
the Mediterranean.
(2) All unarmed steam-vessels are in the Mediterranean.
(3) All steam-vessels not of the Mediterranean are armed.
232 EOUATIONAL LOGIC. [CHAP.
(4) The vessels of the Mediterranean consist of all
unarmed steam-vessels, any number of armed
steam-vessels, and any number of unarmed vessels
without steam. (Boole, The Calculus of Logic,
Cambridge and Dublin Mathematical Journal^
1848, vol. iii. pp. 1991.)
25. How would you otherwise describe the class of things
which are excluded from the class of white, malleable,
metals ?
26. Show that the description of the class of things
which are not (either A, or if not A then both B and C), is
as follows either not-A and not-C, or if it be C then both
not-A and not-B.
27. How do any two of the three equations A = B
B = C, C = A, differ in logical force from the third ?
28. Frame a sorites with one premise negative and one
particular, and represent it equationally.
29. Contrast the logical force of each of the proposi
tions A = AB -I- AC -I- AD .|. . . and A = ABCD . .,
with that of the group of propositions A = AB, A = AC,
A = AD, etc. ; point out, moreover, which can be inferred
from which.
30. Show that, under the condition of our Criterion of
Logical Consistency (p. 181) the assertion that there are no
such things as fresh-water foraminifera, involves the asser
tion that there are foraminifera which are not fresh-water
beings, and fresh-water beings which are not foraminifera,
but leaves doubtful the occurrence of things which are
neither fresh-water beings nor foraminifera.
31. From the premises, All gasteropods are mollusca,
and no mollusca are vertebrates, obtain descriptions of
the classes gasteropods and invertebrates.
xxm.] EXERCISES. 233
32. Eloquence should contain both what is agreeable,
and what is real ; but what is agreeable should be real
(Pascal, Pensees). Represent the above symbolically,
putting A = component of eloquent speech, B = agreeable,
C = real.
33. Assuming it to be known that all mammals have red
blood corpuscles, and that they also have vertebrae, invent
five or six other distinct assertions which you might make
about mammals, the possession of red blood corpuscles, and
the possession of vertebrae, including of course the negatives
of these terms, without coming into logical conflict with the
known relations of the terms as above stated.
34. How would you otherwise describe the class of things
which are excluded from the class of non-crystalline solids
which are either non-metallic non-conductors, or else metal
lic conductors, and which are moreover either brittle and
in that case useless for telegraphy, or else malleable and in
that case useful for telegraphy ?
35. Compare the following propositions :
(1) X is Y.
(2) X is Y and is in some cases Z, and in some cases
not Z.
By the law of excluded middle we know that X must be
either Z or not Z. Is then the sentence (i) precisely
identical in logical force with (2)? Compare now the
following definitions :
(3) A right-angled triangle is that which has a right
angle.
(4) A right-angled triangle is that which has a right angle,
and of which two sides are or are not equal.
Are these definitions precisely identical in logical force ?
[C-]
234 EQUATIONAL LOGIC. [CHAP.
36. What is the difference between saying that sea-water
is drinkable and not scarce, and saying that drinkable sea-
water is not scarce ?
37. If from the premises All rectangles are parallelo
grams, and Parallelograms consist of all four-sided figures
whose opposite sides are parallel, we infer that all rectangles
are parallelograms, being four-sided figures with opposite
sides parallel, how far does this inference fall short of being
equivalent to the premises ?
38. To say that Adam Smith is the father of Political
Economy and a Scotchman is as much as to say that he is
a Scotch father of Political Economy, and that no one but
he can be a father of the science. Give the symbolic proof
of this equivalence.
39. To lay down the condition that what is either A or
else B, is what is both A and B or else both A and C and
vice versd t is to state disjunctively what may be laid down
in two non-disjunctive propositions asserting that A without
B is C and also B must be A.
40. Reduce the two assertions A = Kbc and a = ac to a
single one.
41. Give a good many inferences from the proposition
A = B -|- AC, and also equivalents, distinguishing carefully
between those inferences which are equivalent and those
which are not.
42. Develop symbolically the term Plant (A) with refer
ence to the undermentioned terms (B, C, D, E, F), under
the conditions that acotyledonous (b) plants are flowerless ;
(c) monocotyledonous (D) plants are parallel-leaved (E);
dicotyledonous (F) plants are not parallel-leaved ; and
xxiii.] EXERCISES. 235
every plant is either acotyledonous, monocotyledonous, or
dicotyledonous, but one only of these alternatives.
43. Completely classify triangles under the following
conditions
(1) Equilateral triangles are isosceles.
(2) Scalene triangles are not isosceles.
(3) Obtuse-angled triangles are not right-angled.
(4) Acute-angled triangles have three acute angles.
(5) Obtuse-angled triangles have not three acute angles.
(6) Equilateral triangles are not right-angled.
What other conditions must be added to comply with the
results of geometrical science ?
44. Among plane figures the circle is the only curve of
equal curvature. Show that this is the same as to assert
that a plane figure must either be a curve of equal curva
ture, in which case it is also a circle, or else, not a circle
and then not a curve of equal curvature.
45. Which of the following propositions are equivalent to
the first in the list ?
(1) Crystallised carbon is not a conductor.
(2) Carbon which conducts is not crystallised.
(3) Conducting crystallised substance is not carbon.
(4) Conductors are either not carbon or not crystallised
substances.
(5) Carbon is either not a conductor or not crystallised.
(6) Conductors which are not carbon are crystallised.
(7) Crystals are either non-conductors or not composed
of carbon.
(8) Crystallised conductors are carbon.
46. Prove that any set of exclusive alternatives combined
with part of that set produces only that part.
236 EQUATIONAL LOGIC. [CHAP.
47. Show that the conclusions of Celarent, Cesare,
Camestres, and Camenes give in each case only half the
information contained in the premises.
48. Verify by various trials the statement that no inference
by substitution within a group of propositions can negative
combinations not negatived by the group of premises.
49. Show that Cesare and Camestres belong to the same
type of assertion as Barbara and Celarent.
50. Assign the premises of the following moods of the
Syllogism to their proper types of assertion : Darapti,
Bramantip, Camenes.
51. Prove that any proposition which is contradictory to
common salt = sodium chloride, can be inferred, so far
as it is contradictory, from the assertion common salt =
what is not sodium chloride.
52. Does it or does it not follow that any proposition of
the ;;/th type (see pp. 221-2) will always be equally con
tradictory to one of the wth type ?
53. Refer to Boole s Laws of Thought, pp. 146149,
and taking the premises of the complex problem there
solved to be expressed in our system as follows :
(1) ac = <wE(B// -I- D);
(2) AD^ = AD^(BC -I- be);
(3) A (B -I- E) = G/ [ cD ;
work out the consistent combinations, and infer descriptions
of the classes B, AC, AO>, D, e, AB, AB^, ab, AE, ACE,
BD, DE, De, C, CD, etc. Verify by showing that D and e
multiplied together give De and so forth.
54. If Brown asserts that all metals are reputed elements,
and that all reputed elements will be ultimately decomposed,
whereas Robinson holds that all metals are reputed elements
XXIIL] EXERCISES. 237
which will be ultimately decomposed, what is the exact
amount of logical difference between them ?
55. Compare the logical force of all the following pro
positions, and point out which pairs are equivalent, and
which may be inferred from other ones.
(1) A square is an equal-sided rectangle.
(2) What is not equal-sided is not square.
(3) What is not square is not equal-sided.
(4) Equal-sided rectangles are squares.
(5) No rectangle which is not equal-sided is square.
(6) A square can be neither unequal-sided nor anything
but a rectangle.
(7) An unequal-sided square does not exist.
56. Taking letters to represent qualities thus : A =
having metallic lustre ; B = malleable ; C = heavier than
water ; D = white coloured ; E = fusible with difficulty ;
F = conducting electricity; form descriptions of each of
the metals gold, silver, platinum, copper, iron, lead, tin,
zinc, antimony, sodium, and potassium, and then exhibit
the extension of the following classes : AB ; BC ; BCD ;
BCF; A; be; B//; and so forth.
57. Express symbolically the following classes of things
(1) Hard, wet, black, round, heavy, stone.
(2) Thing which is hard, wet, either black or red, but
not round, and either heavy or not heavy.
(3) Thing which is either not hard, or not wet, or not a
stone, but is either black and then round, or heavy
and then a stone.
58. Referring to the Principles of Science (pp. 75 6 ; ist
ed. vol. i. p. 90), develop all the alternatives of A as limited
by the description
A = AB {C -I- D (E -I- F)}
238 EQUATIONAL LOGIC. [CHAP.
and infer descriptions of the following terms, Ace, Acf, AB<rD.
(See De Morgan, Formal Logic, p. 1 1 6 ; Third Memoir on
the Syllogism, p. 12 in the Camb. Phil. Trans., vol. x.)
59. Represent this argument symbolically: A straight
line can cut a circle in two points, and similarly an ellipse,
and a hyperbola ; but these are all the possible kinds of
conic sections ; therefore a straight line can cut any conic
section in two points.
60. It being understood (i) that only the congenitally
deaf are mute ; (2) that an uneducated deaf person is mute,
but uses signs ; (3) that an educated deaf person is not
mute, and does not use signs : express these conditions
symbolically and describe the classes of persons who are
deaf; mute; deaf-mutes; educated persons, etc.
6 1. Show how by the process of substitution alone to
sum up into one disjunctive proposition the assertion that
John is mortal ; Thomas is mortal ; William is mortal.
62. Prove that the premises of syllogisms in the moods
Darapti and Felapton can be expressed in the form of a
single non-disjunctive proposition, and assign its type. Show
also that this is not the case with the moods of the other
three figures.
63. Prove that the following propositions or groups of
propositions involve self-contradiction :
(i) A = B -I- b.
/ \ (B = AB
^ i = A*.
(3) A = AB ; B = BC ; C = aC.
64. Analyse the force of Hamilton s form of proposition,
* Some A is not some A, putting for some and some
respectively the letter terms P and Q.
EXERCISES. 239
65. What does the assertion Some things are neither A
nor B tell us about things which are not-A ?
66. How far do the conclusions of the syllogisms in
Darapti, Felapton, Bramantip, Camenes, and Fesapo, as
deduced on p. 188, respectively fall short of containing all
the information given in the premises ?
67. Show that C = AC -|- BC is equivalent to the two
propositions, AB = AB<: and ab = abc. Name the type.
68. To say that whatever is devoid of the properties of
A must have those either of B or of D, or else be devoid
of those of C, is the same as to say that what is devoid of
the properties of B and D, but possesses those of C, must
have A. Prove this.
69. What statement or statements must be added to the
proposition, What is not a square is either not equal-sided
or not a rectangle, in order to make the assertions in the
whole equivalent to the definition of a square that it is an
equal-sided rectangle ?
70. What is the difference between the assertion A =
ABC and the pair of assertions b ab, and c - be ?
71. Prove that from one of the propositions, A = ABC,
and AB = ABC, we can infer the other, but not vice
versa, and point out which is the one which can be so
inferred.
72. Give three logical equivalents to the proposition,
ACD = ACD.
73. Demonstrate the equivalence of A = AB [ AC with
Al> = AC, and with Ac = AB<*.
74. Show how by substitution alone to obtain, A = AB
from A = ABCD ; also obtain A = AC" and A = AD.
(Principles of Science, p. 58 ; ist ed. vol. i. p. 69.)
240 EQUATIONAL LOGIC. [CHAP.
7 5. Verify the statement that any set of alternative terms
combined with the same set, reproduces that set that is to
say, show that AA = A when for A we substitute any one
of the followin terms :
ABC -I-0BC -I- abc\
AB^ ! AC -I- tfBC.
76. Show by trial that if in any pair of logically equivalent
assertions such as A = Kb and B = #B, we substitute
for A and B any logical expressions, such as CD for A,
and CE for B, and their negatives in like manner for the
negatives of A and B, we always obtain new equivalent
assertions.
77. As a further example of equivalent assertions take
the following pair of propositions :
f AB = ABC,
{ Ac = Me,
and substitute as follows :
A = PQ, B = Qr, C - PR.
78. Express a - ab and hb - A^C in the form of a
single disjunctive proposition. To what type does it
belong ?
79. Express equationally De Morgan s forms of propo
sition (Formal Logic, p. 62).
(1) Everything is either A or B ;
(2) Some things are neither As nor Bs.
80. Verify the identical equations
A-l- B = A -I- #B;
AB - A (a -I- B)
A -I- BC = (A -I- B) (A -I- C).
xxiii.] EXERCISES. 241
8 1. Verify the following equivalences as transcribed from
De Morgan s Syllabus, p. 42 :
f A = (B ! C) D, J A = (B -I- C) (D -|- E),
\ a be -|- </; ( = be -\- de ;
f A = BC -|. D, j- A = B -I- C (D -|- E),
\a = (H-^; U = (<M-<fc);
J A = (B ! CD) (E ] FG), J A = B i C -|- JD,
\a-bc -|- &/ i </!* <fj I = &*
82. State all the propositions involving only the terms
named which can be inferred from the equation, Stone =
rock ; and all the propositions which are equivalent to this
one, Stone = stone-rock.
83. Show how by the mere process of substitution you
can draw the proposition A = AD from the three propo
sitions A = AB, B =- BC, and C = CD.
84. What propositions added to A = AD are exactly
equivalent in meaning to A = AB, B = BC, and C =
CD jointly?
85. If both A and B have the property C, but A never
occurs where D is, and B never occurs where D is absent,
what is your description of the class of things which are
devoid of the property C ?
86. The proposition A = A (B -|- C) being equivalent to
b - ab -|- A^C, verify this truth by showing that it holds
good when for A we substitute the term P^ -|-/Q, for B
the term QRS, and for C the term q&s.
87. If a person were, correctly or incorrectly, to define
Members of Parliament (including Lords and Commons) as
either peers not chosen by election, or else not-peers chosen
by election, that is as much as to assert both that all members
are non-elected peers and elected non-peers, as well as that
R
242 EOUATIONAL LOGIC. [CHAP.
all who are not members comprise the two classes of persons
who are neither peers nor elected persons, and those who
being peers have been elected but cannot sit.
88. It is not correct to say that because what is not A,
but is B, is also C, therefore everything that is both B and
C is A; but what further conditions may be laid down
about the same things which will render these propositions
convertible ?
89. Into what other equivalent forms might we throw the
joint statements that Venus is a minor planet, and minor
planets are all large bodies revolving round the sun in
slightly elliptic orbits within the earth s orbit ?
90. If B is always found to coexist with A, except when
X is Y (which it commonly, though not always, is), and if,
even in the few cases where X is not Y, C is never found
absent without B being absent also, can you make any
other assertion about C ? [R.]
91. If whenever X is present, Z is not absent, and some
times when Y is absent, X is present, but if it cannot be
said that the absence of X determines anything about either
Y or Z, can anything be determined as between Z and Y ?
[R.]
92. If it is false that the attribute B is ever found co
existing with A, and not less false that the attribute C is
sometimes found absent from A, can you assert anything
about B in terms of C ? [c.]
93. Referring to the Elementary Lessons in Logic, p. 196,
from the premises there given (A = AB -|- AC, B = BD,
C = CD), derive descriptions of the terms BC, a, b, d.
94. From the important problem of Boole, described on
p. 197 of the same lesson, with the premises A = CD,
BC = BD, derive descriptions of the terms BC, bC,
B, *, d.
xxm.] EXERCISES. 243
95. In reference to this last named problem, examine
each of the following assertions, and ascertain which of
them are consistent with the premises A CD, BC = BD
(1) ac = acD. (4) cd acd.
(2) a = acd. (5) Kb = A^CD.
(3) ACD = ABCD. (6) abc = abed.
96. The premises AB = ABC, A = AB, and A = A^,
involve self-contradiction. What is the least alteration which
will remove this contradiction ?
97. If AB = CD, what is the description of BD, of bd
and
98. What must we add to the premises, All As are Bs
and all Bs are Cs, in order that we may establish the rela
tion that what is not A is not C ?
99. Verify the assertion (Principles of Science, p. 141 ;
first edit. vol. i. p. 162) that the six following propositions
are all of exactly the same logical meaning :
A = BC -I- be a = bC -|- Re.
B = AC -I- ac b = Ac -|- aC.
C = AB -I- ab c = aR >\- Kb.
100. Write out five similar logical equivalents of the pro
position r PQ ! pq.
1 01. Prove that ab = abC is equivalent to ac = aeR,
and AB = AC to A = ABC -|- Kbc.
102. How may the condition A [ B = ACD ! BCD
be expressed in four non-disjunctive equations ?
103. Verify the equivalence of M = M and N = Nm,
244 EQUATIONAL LOGIC. [CHAP.
when for M and N we substitute successively the following
pairs of values :
f M = A, f M = M,
(I) IN = ABC. (2) \N =cD.
, x ( M = ACD -I- AMD -I-
(3) \ N - D -I-
104. Express each of the following propositions equa-
tionally in a series of non-disjunctive propositions :
(1) Either the king is dead, or he is now on the march.
(2) Either compression or expansion will produce either
heat or cold in a solid body.
(3) Kb -I- bC = G/ -\>cD.
(4) AB -I- AC - (AB -I- AC) (G/-I- cD).
105. In problem 20 (chap. xxi. p. 194) what description
should we obtain of the classes c t those who do not take
snuff, and d, those who do not use tobacco, respectively
under the several conditions (i), (2) and (3), with (4)?
1 06. In problem 29, pp. 200-1, draw descriptions of
the classes At, ab y and d).
107. Represent symbolically the logical import of the
sentence : If it be erroneous to suppose that all certainty
is mathematical, it is equally an error to imagine that all
which is mathematical is certain.
1 08. Represent equationally the logical import of this
extract from the Oath of Supremacy : No foreign prince,
prelate, person, state, or potentate, hath any jurisdiction,
power, superiority, pre-eminence, or authority, ecclesiastical
or spiritual, within this realm. Observe especially how
far the alternatives are or are not mutually exclusive.
xxiii.] EXERCISES. 245
109. Take the following syllogism in Datisi :
All men are some 1 mortals ;
Some" men are some" 1 fools ;
Therefore, Some iv fools are some v mortals;
and analyse equationally the meanings of the word some
as it occurs five times. Show which of the somes if any
are exactly equivalent. Compare the result with the
remarkably acute analysis of this mood given by Shedden,
in his Elements of Logic, 1864, PP- I 3i-2.
no. If some Xs are Ys, and for every X there is some
thing neither Y nor Z, prove that some things are neither
Xs nor Zs. [DE MORGAN.]
in. Solve equationally Boole s example of analysis of
Clarke s argument (see Laws of Thought, Chap, xiv.)
The premises may be thus stated :
/ ABD = 0. B/ = O.
\ KM =0. AF - O.
CDE = 0. A* = O.
112. Show that every equational proposition whatsoever,
the members of which are represented by X and Y in
X = Y, may be decomposed into two propositions of the
forms X = XY and Y = XY, which will not however
always differ. Show also that the operation when repeated
gives no new result.
113. Take the definition Ice = Frozen Water, and throw
it into equivalent propositions of the following forms :
(1) One disjunctive proposition.
(2) Two non-disjunctive propositions.
(3) One disjunctive and one non-disjunctive.
(4) Two disjunctives and one non-disjunctive.
(5) One disjunctive and two non-disjunctives.
246 EQUATIONAL LOGIC. [CHAP.
(6) Three disjunctives.
(7) Four non-disjunctives.
Are these forms exhaustive, or can you frame yet other
equivalent forms.
114. How many and what non-disjunctive propositions
will be equivalent to the single disjunctive, Kb >\> bC =
Cd [ d3 ?
115. Express the proposition AB = C ] D in the form
of two disjunctive and then in three non-disjunctive
propositions.
1 1 6. As an exercise on Chapter XXII., take the
proposition :
Stratified Rocks = Sedimentary Rocks, and discover (i)
one equivalent ; (2) two inferrible ; (3) several partially
inferrible and otherwise consistent ; (4) several consistent,
indifferent, and not inferrible ; (5) two partially inferrible,
partially contradictory ; (6) one partially indifferent, partially
contradictory; (7) one purely contradictory proposition.
117. Treat in the same general manner any of the
following premises :
(1) Blood-vessels = arteries -|- veins.
(2) Either thou or I or both must go with him.
(3) Heat is conveyed either by contact or radiation.
(4) An equation is either integrable or not integrable.
(5) Roger Bacon, an English monk, was the greatest of
mediaeval philosophers.
(6) Those animals which have a brain in connection
with a spinal cord, and they alone, have red
blood corpuscles. [MURPHY.]
1 1 8. Perform an exhaustive analysis of the relations of
the following propositions, comparing each proposition
XXIIL] EXERCISES. 247
with each other in all the fifteen possible combinations, and
ascertaining concerning each pair under which of the seven
heads it falls :
(1) A = BC. (4) a = BC abc.
(2) Kb = Afo (5) ab = ac.
(3) A = A; B = C (6) AB = ABC.
1 1 9. Perform a similar exhaustive analysis of the rela
tions of the following propositions :
(1) Mercury = liquid, metal.
(2) Not-mercury is not liquid.
(3) Not-metal is either not-mercury or not-liquid.
(4) Mercury is a metal and is liquid.
(5) Liquid is either mercury or not-metal.
(6) Not-liquid is either not-mercury or metal.
(7) Not-mercury is either not-liquid or not-metal.
The eight propositions in question 45 or the seven in 55
of this chapter may be similarly analysed.
120. Analyse this argument: As we can only doubt
through consciousness, to doubt of consciousness is to
doubt of consciousness by consciousness.
121. Illustrate the principle that the relations of logical
symbols are independent of space -relations. (See Prin
ciples of Science, first ed. vol. i. pp. 39-42, 444; vol. ii.
p. 469; new edition, pp. 32~35> 3^3, 769-)
122. Show that if certain premises involving three terms
leave five or more combinations unnegatived, the premises
in question must be self-consistent
123. From the point of view of equational logic analyse
248 EOUATIONAL LOGIC. [CHAP. xxm.
the metaphysical wisdom of Coleridge s doctrine of the
syllogism thus expressed (Table Talk, vol. i. p. 207) :
All Syllogistic Logic is i. delusion; 2. /Delusion;
3. Conclusion - } which answer to the Understanding, the
Experience, and the Reason. The first says : " This
ought to be," the second adds: "This is" and the last
pronounces : " This must be so." ;
CHAPTER XXIV.
THE MEASURE OF LOGICAL FORCE.
1. THE combinational analysis of the meaning of asser
tions enables us to establish an almost mathematical system
of measurement of the comparative force of assertions.
Given the number of independent terms involved, that
form of proposition has the least possible force which
negatives only a single combination. Thus with three
terms, a proposition of the form AB = ABC negatives only
the single combination ABr ; but A = ABC negatives three,
and A = BC as many as four combinations. These latter
propositions may be said to have three and four times the
logical force of the first given.
2. I have not yet been able to discover any general laws
regarding this subject of logical force, but many curious
and perhaps important observations may be made. Thus
a great many forms of assertion agree in having the logical
force one-half, that is to say, they negative half the com
binations. Such is the case, the terms being three in
number, with the propositions A = BC ; A = B -|- BC ;
A = Br -|- bC. Indeed, it is very frequently true that any
proposition having no term common to both sides of the
equation negatives half the combinations. This is true of
all propositions of the types A = B, A = BC, and generally
A = BCD . . . Y. But it is not true of the type AB = CD.
250 EQUATIONAL LOGIC. [CHAP.
The appearance of the same term in both members of an
equation always weakens its force ; thus A = ABC has the
force only of f , whereas A = BC has the force i. Again,
A = B -|. C has the force -J-, but A ! B = B -|- C only the
force f .
3. The best ostensive instance of logical power is found
in a form of proposition which embraces the greatest in
tension in one member with the greatest extension in the
other. This kind of assertion has the general form
ABC . . . = P .< Q -I- R !...; and as the terms increase
the logical force approaches indefinitely to unity. Thus
while A = B ! C has the value , AB = C ] D has that
of 10 out of 1 6, and A -|- B = CDE that of 22 out of 32.
A few other observations on this subject are thrown into
the form of questions :
4. Show that the logical force of n equations of the
form A = B, B = C . . . . is i -
5. Prove that a single proposition of the type ABC ....
= P .[. Q ) R ] . . . ., there being in all n independent
letter terms, and no term common to both members, has the
logical force i + ^-^ which approaches indefinitely
to unity as n increases.
6. Can you discover any equation between a single term
and any expression not involving that term which has a
logical force other than one-half?
7. What form of proposition involving only A and B in
one member, and C, D, in the other, has the lowest possible
logical force ?
8. What is the utmost number of combinations of ;/
terms which can be negatived without producing con
tradiction ?
xxiv.] MEASURE OF LOGICAL FORCE. 251
9. What is the utmost number of combinations of four
terms which can be negatived by a proposition involving
only three of them ?
10. What two propositions involving five terms negative
the utmost possible number of combinations, without self-
contradiction ?
11. Show that m successive propositions of the type
A = AB, B = BC . . . ., that is to say, in the form of the
Sorites, leave m + 2 combinations unnegatived, so that the
, . , c . m + 2
logical force is i - m + i
12. Prove that the amount of surplus assertion, or over
lapping of the propositions, in a Sorites as treated in the
last question, increases indefinitely. Investigate the law of
the surplusage.
13. What is the utmost possible logical force as regards
m terms of an equation involving n terms.
CHAPTER XXV.
INDUCTIVE OR INVERSE LOGICAL PROBLEMS.
1. THE direct or deductive process of logical analysis
consists in determining the combinations which are, under
the Laws of Thought^ consistent with assumed conditions.
The Inverse Problem is given certain combinations incon
sistent with conditions, to determine those conditions. As
explained in the Principles of Science (chapter vii.) the
inverse problem is always tentative, and consists in invent
ing laws, and trying whether their results agree with those
before us. An American correspondent, Mr. M. H.
Doolittle, points out that in making trials we should always
pay attention to combinations in proportion to their in
frequency , or solitariness, infrequency being the mark of
deep correlation. The infrequency may be that either of
presence or of absence.
2. The following inductive problems consist of series of
combinations of three terms and their negatives which are
supposed to remain uncontradicted under the condition of a
certain proposition or group of propositions. The student
is requested to discover such propositions, express them
equationally, and then assign them to the proper type in
the table on p. 222. If in any problem the conditions are
self-contradictory the student is to detect the fact.
CHAP, xxv.] INVERSE PROBLEMS. 253
I. IV. VII. IX.
ABC ABC AC
abc aEc kbc
abC #BC
n - aBc abc
Kbc V.
AC VIII. X.
Kbc ABC ABC
Kbc
^BC VI.
aRc KbC
Kbc
abc
3. Assuming each of the following series of combinations
to consist of those excluded or contradicted by certain proposi
tions , assign the propositions which are just sufficient to
exclude them in each problem, express these propositions
equationally, and refer them as in the last question to the
proper type in the table :
I. V. VIII.
ABC ABC ABC
II. hbc
abC VI abc
III. AB IX.
Kbc
abc
VII. Kbc
abC abc
IV.
ABC abC
abc abc abC
254
EOUATIONAL LOGIC.
[CHAP.
4. I now give a series of inductive problems involving
four terms. Each series of combinations consists of those
which remain after the exclusion of such as contradict
certain conditions. Required those conditions. The
problems are ranged somewhat in order of difficulty.
I.
ABCD
abed
II.
ABCD
III.
abCd
abeV
abed
IV.
ABCD
a&CD
abed
VI.
ABG/
KbCd
abed
VII.
ABCD
ABO/
hbcd
aEed
ABC^
V.
It C/t-t*
0BCD
ABCD
aECd
ABC^
VIII.
^CD
AB^
Kbea
abCd
A^CD
aECd
abcD
A^C^
abCd
abed
abed
abcft
abed
IX.
aEeD
aftea
abCd
X.
ABd)
aRcd
abea
XI.
AB^D
abCd
abed
XII.
ABCD
abed
5. I next give a few similar problems involving five or six
terms, as follows :
XXV.]
INVERSE PROBLEMS.
255
I.
III.
V.
ABODE
ABC^E
AB^DE
KBcde
ABfl/E
AB<;D^
Abcde
ACWE
AB^E
aEcde
KbcT>e
K&cde
abcde
abcdE
abCTie
abcde
abCde
II.
IV.
VI.
ABC/fc
ABCDE
ABCDE
aECde
KbCde
ABC^/E
abCde
^BrDE
ABd>
abcDE
abCDe
A^CD^
abcDe
abcdE
^BrDE
abcde
abCde
VII.
VIII.
ABcDef
AECJef
AEcdEf
AbCVef
aBfDef
dBcdEE
aBcdEf
a&CDef
a&CDeY
abcDEF
AB^DEF
A&CVef
AtcDEf
IX.
hbcDef
AfcdEF aQcdEE
AbcdeF ^CDEF
a&CDef
abcDef
abcGEf
dbcdeS
256 EOUATIONAL LOGIC. [CHAP.
6. As the reader who is in possession of the present
volume will have plenty of unanswered inductive problems,
it may be well to give here the answers to the problems of
the like kind which were set in the Principles of Science^
new edition, p. 127. They are as follows :
I. A = AB^- -I- AC ; a = 0BC.
II. A = AC ; a = ciB.
III. A = AC ; db = abc.
IV. A = D ; B = CD -I- cd, or their equivalents
A = D ; a = >c -I- bC.
V. ab = abCd } Kb =
VI. D = E ; bC = CD ; (a ! b) c = abcde.
VII. A =^ = D =e- } B = ^B.
VIII. (Unknown.)
IX. btt = C -I-/; atiE = B^F ; ACF = AC^F ;
X. This example was set by me at haphazard, like
Nos. V. and VI. , that is to say, by merely striking out any
combinations of the logical alphabet which fancy dictated.
Dr. John Hopkinson, F.R.S., has given me the following
rather complex solution
(1) d=abd.
(2) b = b (AF .|. ae).
(3) A/=A/Bd)E.
(4) E = E(B/.|-JACDF).
(5) ^B = ^ABCDF.
(6) abc abcef.
(7) abef = abcef.
Can a simpler answer be discovered ?
xxv.] INVERSE PROBLEMS. 257
7. In the first edition of the Principles of Science, vol. ii.
p. 370, I gave a rather complex problem involving six
terms, the combinations unexcluded being as follow after
coalescence of some alternatives :
ABCDF
AECDef
ABC^/E/
ABd)F bcd E.f
To a request for solutions I received several answers,
mostly correct, but, curiously enough, all differing in the
forms of proposition. The correct ones are given below
as furnishing a remarkable instance of logical equivalence.
First Answer. Second Answer.
C = ABC b = cd
f =T)ef -\-JEf C=ABC
c = Bd) -I- bed D = BD
d = </E/ d = Ef
Third Answer. Fourth Answer.
a ac C = AC
b = ca b = bed
d - E/ d = E/
B = B (C -I- D)
Fifth Answer.
AC = BC
BDE = BDEF
B = C -I- D
d = </E/
s
258 EQUATIONAL LOGIC. [CHAP. xxv.
The third answer, given by Mr. R. B. Hay ward, M.A.,is
in the simplest terms. The propositions inserted as the
fifth answer are those from which I formed the combinations
deductively. The student may prove that any one of these
answers is deducible from any other without descending
explicitly to the combinations ; thus C = AC is the con-
trapositive of a = ac ; B = C ] D is equivalent to
b = cd t and so forth.
CHAPTER XXVI.
ELEMENTS OF NUMERICAL LOGIC.
i . LET a logical term, when enclosed in brackets, acquire
a quantitative meaning, so as to denote the number of
individual objects which possess the qualities connoted by
the logical term. Then (A) - number of objects possess
ing qualities of A, or say, for the sake of brevity, the number
of As.
Every logical equation now gives rise to a corresponding
numerical equation. Sameness of qualities occasions same
ness of numbers. Hence if A = B denotes the identity of
the qualities of A and B, we may conclude that (A) = (B).
It is evident that exactly those objects, and those objects
only, which are comprehended under A must be compre
hended under B. It follows that wherever we can draw an
equation of qualities, we can draw a similar equation of
numbers. Thus, from A = B = C, we infer A = C ; and
similarly from (A) = (B) = (C), meaning the number of
As and Cs are equal to the number of Bs, we can infer
(A) = (C). But, curiously enough, this does not apply to
negative propositions and inequalities. For if A = B < D
means that A is identical with B, which differs from D, it
does not follow that
260 NUMERICAL LOGIC. [CHAP.
Two classes of objects may differ in qualities, and yet they
may agree in number.
2. The sign ] being used to stand for the disjunctive
conjunction or, but in an unexclusive sense, it follows that
|- is not identical in meaning with +. It does not follow
from the statement that A is either B or C, that the number
o As is equal to the number of Bs added to the number of
Cs ; some objects, or possibly all, may have been counted
twice in this addition. Thus, if we say An elector is either
an elector for a borough, or for a county, or for a university,
it does not follow that the total number of electors is equal
to the number of borough, county, and university electors
added together ; for some men will be found in two or three
of the classes.
This difficulty, however, is avoided with great ease ; for
we need only develop each alternative into all its possible
subclasses and strike out any subclass which appears more
than once, and then convert into numbers, connected by
the sign of addition. Thus, from A = B -|- C we get
A = BC -I- B^ -I- BC -I- C; but striking out one of the terms
BC as being superfluous, we have A = BC -|- B<: ] bC.
The alternatives are now strictly exclusive, or devoid of
any common part, so that we may draw the numerical
equation
(A) = (BC) + (Br) + (JQ.
Thus, if
A = elector, C = county elector,
B = borough elector, D = university elector,
we may from the proposition A = B -|- C -|- D draw 7 the
numerical equation
(A) = (BCD) + (BCO + (B;D) -f (B*/) + (JCD) +
+ (bcD).
xxvi.] QUESTIONS AND ANSWERS. 261
3. The data of any problem in Numerical Logic will be
of two kinds :
(1) The logical conditions governing the combinations
of certain qualities or classes of things, expressed
in propositions.
(2) The numbers of individuals in certain logical
classes existing under those conditions.
The qucesita of the problem will consist in determining
the numbers of individuals in certain other logical classes
existing under the same logical conditions, so far as such
numbers are rendered determinable by the data. The
usefulness of the method will, indeed, often consist in
showing whether or not the magnitude of a class is deter
mined or not, or in indicating what further hypotheses or
data are required. It will appear, too, that where an exact
result is not determinable we may yet assign limits within
which an unknown quantity must lie.
4. In a certain statistical investigation, among
100 As there are found 45 Bs and 5 3 Cs ; that
is to say, in 45 out of I oo cases where A occurs
B also occurs, and in 5 3 cases C occurs. Sup
pose it to be also known that wherever B is,
C also necessarily exists. It is required to
determine
(1) The number of cases (all being As) where C
exists without B.
(2) The number of cases (all being As) where
neither B nor C exists.
262 NUMERICAL LOGIC. [CHAP.
The data are as follow :
r(A)=ioo ..... (i)
Numerical equations -< (B) =45 ..... (2)
l(C)= 53 ..... (3)
Logical equation . . B = BC.
The logical equation asserts that the class B is identical
with the class BC, which is the true mode of asserting
that all Bs are Cs. Two distinct results follow from this,
namely: ist, that the number of the class BC is identical
with the number of the class B ; and 2d, that there are
no such things as Bs which are not Cs.
The logical equation is thus equivalent to two additional
numerical equations, namely,
(B) - (BC) ....... (4)
(B.) = o ........ (5)
We have now means of solving the problem ; for, by the
Law of Duality,
By (4)
- (B) + (JC).
Thus
53 = 45 + (JQ,
or the required number of Cs is 8.
To obtain the number of AArs, we have
(A) = (ABC) + (ABr) + (AJC) + (Me)
i oo =45 + +8 + (A&r).
Hence
(A&) = 47-
xxvi.] QUESTIONS AND ANSWERS. 263
5. The difference between the numbers of objects
in any two classes whatsoever, is equal to the
difference between the numbers of objects
which are in each class, but excluded from the
other class.
Take (A) and (B) to represent the numbers in any two
classes A and B ; then
(A) - (B) - (AB) + (A) - (AB) - (*B)
= (AJ) - (B).
6. If the number of As be x, of Bs be 7, and of
those Bs which are not As be /, then the
number of As which are not B will be/ + x - y.
Setting down the several logical quantities represented by
/ 4- x - y, we have
(*B) + (AB) + (A*) - (AB) - (*B).
Four terms cross out, leaving only (A) as required.
7. Represent the following argument from
Thomson s Laws of Thought, p. 168 :
Three-fourths of the army were Prussians ;
Three-fourths of the army were slaughtered ;
Therefore some who were slaughtered were
Prussians.
Taking A = members of the army,
B = Prussians,
C = slaughtered,
264 NUMERICAL LOGIC. [CHAP.
the premises are expressed as
(AB) = f (A)
(AC) = f (A).
The number of Prussians slaughtered will be (ABC), of
which the following equation is identically true :
(ABC) = (AB) + (AC) - (A) +
inserting values
(ABC) =
That is to say, the number of Prussians slaughtered was at
least half the army, and exceeds it by a number equal to
the number of men in the army who were neither Prussians
nor slaughtered.
8. If the number of As which are Bs is/, and the
number of Bs which are Cs is q y what do we
know concerning the number of As which are
Cs?
We have the following self-evident equations :
AC - ABC + AC
= ABC + AB;- + ABC + tfBC -f
+ dEc - B
= AB + BC - B + AC +
Inserting the values given, we get
AC =/ + q - B + AC +
We see that the data are quite insufficient for determining
the number of As which are Cs. They may be anything
from zero up to the whole number of As or Cs. To make
the question determinate we need also the number of Bs,
xxvi.] QUESTIONS AND ANSWERS. 265
as well as the number of ACs, which are not Bs, and the
number of Bs which are neither A nor C.
9- Most Bs are As ..... (i)
Most Bs are Cs ..... (2)
Therefore, Some Cs are As ..... (3)
The above argument is a celebrated one, proposed by
De Morgan (Formal Logic, p. 163), and discussed by Boole
(Trans, of the Cambridge Philosophical Society, vol. xi. part
ii. p. i) and others. Regarded as an ordinary Aristotelian
pseudo-syllogism, it is subject to the fallacy of undistributed
middle, since the proposition Most Bs are As/ must be
counted as a particular affirmative. The pseudo-mood is
accordingly 1 1 1 in the third figure. Nevertheless the force
of the argument is pretty obvious and may thus be analysed.
The mark of quantity, most, of course means more than
half, and is one of the few quantitative expressions used in
ordinary language. We can easily represent the two pre
mises in the form
- ....... (i)
(BC) = i(B) + / ....... (2)
To deduce the conclusion, we must add these equations
together, thus,
(AB) + (BC) - (B) + w + w .
Developing the logical terms on each side, we have
(ABC) + (ABc) + (ABC) + (^BC) - (ABC) + (AB*)
+ (0BC) + (B<r) 4- w + w .
Subtracting the common terms, there remains
(ABC) = w + w + (aBc).
266 NUMERICAL LOGIC. [CHAP.
The meaning is, that there must be some Cs which are
As, amounting to at least the sum of the quantities w and
w , the unknown excesses beyond half the Bs which are As
and Cs. The number (aBc) is wholly undetermined by the
premises, but it cannot be negative. Thus w + w is the
lower limit of (ABC).
IO. (i) For every Z there is an X which is Y ;
(2) Some Zs are not Ys. What inferences can
be drawn ?
This general problem given by De Morgan in his Syllabus
(p. 29, art. 85), and in some other parts of his writings,
would thus be represented in my formulas, which differ
essentially, however, from those of De Morgan.
The premises are
(XY)-(Z) + i (i)
in which m would be zero if De Morgan meant that there
are not more Xs which are Ys than there are Zs, but just an
equal number
(Zy) = (2)
where n is some positive number.
Developing (i) we get
(3) (XYZ) + (XYz) - (XYZ) + (XyZ) + (xYZ)
+ (xyZ) + ;//.
Striking out the common term and adding (Xyz) to both
sides, we have for the number of Xs which are not Zs
(Xz) = m +n+ (Xyz) + (xYZ).
xxvi.] QUESTIONS AND ANSWERS. 267
Again, after striking out the common term, equation (3)
reduces to
(XYz) = (XyZ) + (xZ) + m,
which gives as the number of Zs which are not Xs
(xZ) = (XYz) - (XyZ) - ;;/.
The student should compare these results with those of the
less general problem given in the Principles of Science, new
edition, p. 169 ; first edition, vol. i. p. 191, and also with
De Morgan s results expressed in a totally different kind of
notation.
II. If m or more Xs are Ys, and n or more Ys
are Zs, what do we know about the number of
Xs which are therefore Zs ?
This question represents one case of the numerically
definite syllogism as treated by De Morgan (Syllabus, p. 27).
Taking X, Y, and Z to be the three terms of the syllogism,
he adopts the following notation :
u - whole number of individuals in the universe of
the problem.
x = number of Xs.
y = number of Ys.
z number of Zs.
Making m denote any positive number, wXY means, in
De Morgan s system, that m or more Xs are Ys. Similarly
;?YZ means that n or more Ys are Zs. Smaller Roman
letters denote the negatives of the larger ones. Thus wXy
means that m or more Xs are not Ys, and so on.
From the two premises wXY and YZ, De Morgan draws
the conclusion (m + n - jy)XZ. Let us consider what
268 NUMERICAL LOGIC. [CHAP.
results are given by our notation. The premises may be
represented by the equations
(XY) = m + m (YZ) = n + ,
where m and n are the same quantities as in De Morgan s
system, and m and n two unknown but positive quantities,
indicating that the number of XYs is m or more, arid the
number of YZs is n or more.
The possible combinations of the three terms X, Y, Z, and
their negatives are eight in number, and these all together
constitute the universe, of which the number is u. The
problem is at once seen to be indeterminate in reality ; for
there are eight classes of which the numbers have to be
determined, and there are only six known quantities, namely,
z/, x, y, z, m, and , by which to determine them. Accord
ingly we find that De Morgan s conclusion, though not
absolutely erroneous, has little or no meaning. From the
premises he infers that (m + n - y) or more Xs are Zs.
Now
m + n-y= (XY) + (YZ) - (Y) - ; - n
= (XYZ) - (xYz) - m - n .
Thus De Morgan represents the number of the whole class,
XZ, by a quantity indefinitely less than its own part, XYZ.
It is quite true that if the second side (XYZ) - (xYz)
- m - n of this equation has value, there must be at
least this number of Xs which are Zs ; but as (xYz)
may exceed (XYZ) in any degree, this may give zero or a
negative result, while there is really a large number of XZs.
The true and complete expression for the number of XZs
is found as follows :
(XZ) = (XYZ) + (XyZ)
- (XYZ) + (XYz) + (XYZ) + (xYZ) - (Y) + (XyZ)
+ (xYz) = /// + m + n + n - y + (XyZ) + (xYz).
xxvi.] QUESTIONS AND ANSWERS. 269
Among these seven quantities, only ;, ;z, and y are definitely
known. The two m and ri are two indefinite quantities,
expressing the uncertainty in the number of XYs and
YZs, while there are two other unknown quantities, the
numbers of XyZs and xYzs arising in the solution of the
problem.
12. If m or more Xs are Ys, and n or more Ys
are Zs, what do we know about the number of
not-Xs which are not-Zs ?
From the same two premises as in the last problem,
namely
mXY and YZ,
De Morgan draws the conclusion
(m + n 4- u - x - y - z)xz
that is to say, the number of not-Xs which are not-Zs is
the quantity in the brackets or more. This conclusion is
equivalent to that in the preceding problem.
To prove this result it is requisite to develop all the
combinations numbered in each of the quantities ;//, n t u,
x, y, z ; there are twenty-six terms in all which the reader
may readily work out. Giving them the signs indicated by
De Morgan, and striking out pairs of positive and negative
terms, we find only two combinations left, together with
m and n t which terms are used, as in the last problem, to
express the fact that De Morgan s proposition ;;zXY is not
really definite, but means that m or more, that is m or
(/ + m } Xs are Ys. We thus obtain
(xy) - (xyz) - (XyZ) - m - n
in which the term (XyZ) is wholly undetermined. Thus we
find that De Morgan s method gives us as the value of (xy)
270 NUMERICAL LOGIC. [CHAP.
a part of itself (xyz), diminished by three unknown quanti
ties. The number (xz) may accordingly be of any magni
tude, while the lower limit assigned to it by De Morgan is
zero, or even negative. The problem is in fact a wholly
indeterminate one, and De Morgan s solution is illusory.
Similar remarks may be made concerning other conclu
sions which De Morgan draws. Thus, from ;;/Xy and nYz
(mXs or more are not Ys, and ;*Ys or more are Zs) he infers
(m + n x) xZ and (m -\-n-z) Xz.
But it will be found by analysis that the first of these results
has the following meaning :
(xZ) ^ (xYZ - (XYz) ;
that is to say, the lower limit of the class xZ is a part of
itself, xYZ, diminished by the number of another class
XYz of unknown magnitude.
13. If the fractions a and fi of the Ys be severally
As and Bs, and if a + {3 be greater than unity,
it follows that some As are Bs.
[Cambridge Phil. Trans, vol. x. part i. p. 8.]
In his third memoir on the Syllogism De Morgan gives
the above as a very general statement of the conditions of
valid mediate inference. He remarks that the logician,
that is to say, the ordinary Aristotelian logician, demands
a = i or ft = i, or both; he can then infer. This
represents the condition of a distributed middle term.
The numerically definite conditions are readily represented
as follow :
The premises are a . (Y) = (AY).
ft . (Y) = (BY).
xxvi.] QUESTIONS AND ANSWERS. 271
Hence
(a + /?)()= (AY) + (BY)
= (ABY) + (AY) + (ABY) + (*BY),
(a + p) (Y) - (Y) - (ABY) - (abV),
or
(ABY) = ( a + j8 - i) (Y) + (ab\).
We learn that the number of AYs which are Bs is the
fraction ( + ft - i ) of the Ys, together with the undeter
mined number (0Y), which cannot be negative. But,
according to the conditions, a + (3 is greater than unity ;
hence the second side of the equation must have a positive
value. Not only will there be (a + /3 - i) As which are
Bs, but this is merely the lowest limit, and there will be
as many more as there are units in the number of 0Ys.
If we distribute the middle term Y once, by making
a = i , we have
(ABY) = /? . (Y) + o.
The term (abY) of course vanishes because the whole of
the Ys are As. Again, if /3 = i ? we have
(ABY) = a . (Y).
If both a and /3 become unity, then
(ABY) - (Y).
It must be carefully noted however that these results do not
show the whole number of As which are Bs, but only those
which are so within the sphere of the term Y. Nothing
has been said about the combinations of not-Y, which are
quite unlimited by the conditions of the problem.
14. If A occurs in a larger proportion of the
cases where B is than of the cases where B is
272 NUMERICAL LOGIC. [CHAP.
not, then will B also occur in a larger propor
tion of the cases where A is than of the cases
where A is not.
This general proposition is asserted in J. S. Mill s chapter
On Chance and its Elimination, but is not proved by
Mill. (System of Logic, Book III., chapter xvii. section 2,
adfinem; fifth edition, vol. ii. p. 54.) I do not remember
seeing any proof of it given elsewhere, and it is not to my
mind self-evident. The following, however, is a proof of its
truth, and is the shortest proof I have been able to find.
The condition of the problem may be expressed in the
inequality
(AB) : (B) > (A*) : (b\
or reciprocally in the inequality
(B) : (AB) < (*) : (M).
Subtracting unity from each side, and simplifying, we have
(0B) : (AB) < (ab) : (Alt).
Multiplying each side of this inequality by (Ab) : (B) we
obtain
(Ab) : (AB) < (ab) : (aB).
Restoring unity to each side, and simplifying
(A
or reciprocally
which expresses the result to be proved, namely, that B
occurs in a larger proportion of the cases where A is than
of the cases where A is not.
15. In a company of r individuals, p have coats
and q have waistcoats. Determine some other
relations between them.
xxvi.] QUESTIONS AND ANSWERS. 273
Boole treats this problem in the fourth page of his
Memoir On Propositions numerically definite (Cambridge
Philosophical Transactions, vol. xi. part ii.). Taking i to
represent the company which is the universe of the propo
sition, x the class possessing coats, y the class possessing
waistcoats, and using the letter N, according to Boole s
notation, as equivalent to the words number of,
p = NX, q = Ny, r=Ni,
he finds, as we have found in a preceding page (p. 2 6 4, No. 7)
Nxy = p + $ - r + Ni x i y.
Ni x i y = r p q + Nxy.
He proceeds, * Again, let us form the equation
2 P ~ $ ~ r = 2 N# - Ny - N i .
= N (2x - y - i)
= N (xi y 2yi x i xi -y)
= NX i - y - 2Ny i-x-Ni-xi-y.
From which we have
NX i y 2p q r-\- 2Ny i - x + N i x i y.
Hence we might deduce that the number who had coats
but not waistcoats would exceed the number 2p q r
by twice the number who had waistcoats without coats
together with the number who had neither coats nor waist
coats. This is not, indeed, the simplest result with reference
to the class in question, but it is a correct one.
The student is requested to verify this result.
On going over this paper of Boole s again, it becomes
apparent to my mind that his method is identical with
that developed in this chapter and in my previous paper
T
274 NUMERICAL LQGIC. [CHAP.
on the same subject (Memoirs of the Manchester Literary
and Philosophical Society, Third Series, vol. iv. p. 330,
Session, 186970), written with a knowledge, as stated on
p. 331, of Boole s publication on the subject.
16. Can we represent a syllogism in the extensive
form by means of numerical symbols ?
In a very interesting and remarkable paper read to the
Belfast Philosophical Society in 1875, Mr. Joseph John
Murphy has given a kind of numerical notation for the
syllogism. He has since printed a more condensed and
matured account of his views in Mind, January, 1877.
Taking the syllogism Chlorine is one of the class of
imperfect gases ; imperfect gases are part of the class of
substances freely soluble in water ; therefore, chlorine is
one of the class of substances freely soluble in water he
assumes the symbols
x Chlorine, y = Imperfect gases,
z = substances freely soluble in water.
He expresses the first premise in the form
y*x +P,
p being a positive numerical quantity indicating that there
are other things besides chlorine in the class of imperfect
gases. The second premise takes the form
z=y + f,
similarly indicating that besides imperfect gases there are
q things in the class of substances freely soluble in water.
Substitution gives z = x +p + q,
which would seem to prove that besides chlorine (x) there
are p + q things in the class of substances freely soluble in
water.
xxvi.] QUESTIONS AND ANSWERS. 275
The student who wishes to master the difficulties of the
modern logical views should read these papers with great care.
Space does not admit of my arguing the matter out at full
length, and I can therefore only briefly express my objec
tions to Mr. Murphy s views as follows : His equations are
equations in extension, and, with his use of + and - , they
can only hold true when his terms are numerical quantities.
Under this assumption his equations show with perfect
correctness the numbers of certain classes; but they are not
therefore equivalent to syllogisms. Because z = x + p + q,
we learn that the number z exceeds x by p + #, but it does
not therefore follow that chlorine belongs to the class of
substances represented by z. In short, as I have pointed
out at the beginning of this chapter (p. 259), from logical
equations arithmetical ones follow, but not vice versa. (See
also Principles of Science, p. 171 ; first edit. vol. i. p. 193.)
I hold, therefore, that Mr. Murphy s forms are not really
representations of syllogisms ; but at the same time I am
quite willing to admit that this is a question never yet
settled and demanding further investigation. It is very
remarkable that Hallam inserted in his History of Literature
(ed. 1839, vol. iii. pp. 287-8) a long note containing a
theory of the syllogism somewhat similar to that of Mr.
Murphy, but which has hitherto remained unknown to Mr.
Murphy and apparently to all other logical writers.
CHAPTER XXVII.
PROBLEMS IN NUMERICAL LOGIC.
1. IF from the number of members of Parliament we
subtract the number of them who are not military men, we
get the same result as if from the whole number of military
men we subtract the number of them who are not members
of Parliament. Prove this.
2. In a company of x individuals it is discovered that y
are Cambridge men, and z are lawyers. Find an expression
for the number of Cambridge men in the company who are
lawyers, and assign its greatest and least possible values.
[BOOLE.]
3. Prove that in any population the difference between
the number of females and the number of minors is equal
to the difference between the number of females who are
not minors, and of minors who are not females.
4. Show that if to the number of metals which are red,
we add the number which are brittle, the sum is equal to
that of the whole number of metals after addition of the
number of metals which are both red and brittle, and after
subtraction of the number of metals which are neither red
nor brittle.
5. What is the value of the following expression
(A)-(AB)-(AC)?
CHAP, xxvii.] PROBLEMS. 277
6. Prove that the number of quadrupeds in the world
added to the number of beings not quadrupeds which
possess stomachs is equal to the whole number of things
having stomachs together with the number of things not
having stomachs which are quadrupeds.
7. If x and y be respectively the numbers of things which
are X and Y, while m is the whole number which are both
X and Y, and n the number which are either X alone
or Y alone, what is the relation between m + n and
x+y?
8. Let u be the whole number of things under con
sideration, x the number which are A, and y the number
which are B ; then if m be the number of things which are
both A and B, show that m + u x y is the number
which are neither A nor B.
9. Taking each logical term to represent the number of
things included in its class, verify the following equa
tions :
(A - AB) (A - AC) = A - AB - AC + ABC - Kbc.
(A - AB) (A - AC) (A - AD) = A - AB - AC +
ABC - AD + ABD + ACD - ABCD = Kbcd,
10. What is the product of the logical multiplication of
the four factors
(A - AB) (A - AC) (A - AD) (A - AE)?
Give another expression for its value.
11. Show that the following equation is necessarily true :
B + AC + C + A&: = A + C + aEc.
12. What happens in Problem 8 if it be discovered that
the class B does not exist at all ?
13. Find an expression for the difference between (A)
and (B) + (C).
278 NUMERICAL LOGIC. [CHAP.
1 4. What is (a) the minimum percentage of C that must,
and (/?) the maximum that may coincide with B under the
following conditions ?
80 per cent of As coincide with 50 per cent of Bs.
70 per cent of As coincide with 60 per cent of Cs. [D.]
15. If revolutions occur in a larger proportion of govern
ments where the press is under a censorship, than of govern
ments where it is not, then will a censorship of the press be
found in a larger proportion of governments which are sub
ject to frequent revolutions, than of governments which are
not thus subject. [D.]
1 6. If/ per cent of A are B, and q per cent of A are
C, what is the least percentage of A that those individuals
make up which are both B and C ? [D.]
1 7. Show that we cannot tell what percentage of B or of
C the same individuals make up unless we know how much
of B or of C is not A. [D.]
1 8. In the easy case in which all B is A, and all C is A,
find what percentage of B or of C must be made up by the
individuals which are both B and C at once. [D.]
1 9. Prove the following equations :
(Afe) + (AB) = (A) + (ABC) - (AC).
(A + B) - (C + D) = (A + B) (c + C<t) - (C + D)
(a + A6) - AB (Cd + cD) + CD (A + 0B).
20. Prove that the following equation gives a correct
expression for the common part of any three classes -
A, B, C.
(ABC) = (B) + (C) - (A) - (aB) - (aC) + (Ate).
21. In a company consisting of r individuals there were
q in number who knew Latin, and / in number who knew
xxvn.] PROBLEMS. 279
either Latin or French, but not both ; between what limits
is the number of those who knew French confined ?
22. In the last problem prove that the lower limit is the
greatest value in / - q and q - p, and the upper limit, the
least value in 2r - p - q, and/ + q. (See Boole, On Pro
positions Numerically Definite, p. 15.)
23. The student will find many other numerically defi
nite problems in De Morgan s Formal Logic, Chapter VIII.,
and in his Syllabus, pp. 27-30; but in reading De Morgan
it must be carefully remembered that mXY means with him
not that mXs are Ys but that m or more Xs are Ys. His
solutions will sometimes, as shown in the previous chapter,
be found delusive.
24. Verify the following assertion of De Morgan: To
say that mXs are not any one to be found among any lot of
72 Ys is a spurious (that is a self-evident or necessary) pro
position, unless m + n be greater than both x and y, in
which case it is merely equivalent to both of the following,
(/// + n y] Xy, and (m + n - x) Yx, which are equiva
lent to each other.
25. It is found that there are in a certain club of x mem
bers, y London graduates, and z lawyers. What further
numerical data are requisite in order to define the numbers
who are both London graduates and lawyers, and of those
who are neither ?
26. If there are more persons in a town than there are
hairs on any one person s head, then there must be at least
two persons in the town with the same number of hairs on
their heads. Put this theorem into a strict logico-mathe-
matical form. [HERBERT SPENCER.]
27. Demonstrate the theorem in numerical logic given in
the Principles of Science, new edition (only), p. 170.
28. For every man in the house there is a person who
280 NUMERICAL LOGIC. [CHAP. xxvn.
is aged; some of the men are not aged; it follows, and
easily, that some persons in the house are not men ; but
not by any common form of syllogism. (DE MORGAN,
Syllabus, p. 29.) A solution of this problem is given in
Principles of Science, new edition, p. 169.
29. Draw what conclusions you can from the following :
There were some English on board; and though no
passengers were saved from the wreck, and of the ship-
officers, as it happened, only one, yet no Englishman was
lost. [R.]
CHAPTER XXVIII.
THE LOGICAL INDEX.
1. I NOW give what I propose to call the Logical Index,
or, more precisely, the Logical Index of Three Terms. As
however the logical relations of two terms are too simple
to need an index, and those of four terms are vastly too
numerous and complex to admit of exhaustive treatment
at present, the Index of Three Terms is practically the only
one which can be given. It contains, within the space of
four pages, a complete enumeration of all possible purely
logical conditions involving only three distinct terms.
2. Each page contains a double-sided table, forming in
fact two tables. Each such table contains a column of
equational propositions, a column of Roman numerals
showing the type (see p. 222) to which such propositions
belong, a column of consecutive Arabic numbers for sake
of easy reference, and lastly a column of Greek letters,
which supplement the Greek letters a, /?, y, given at the
heads of the columns of propositions. These Greek letters
stand in place of the combinations of the fourth column of
the Logical Alphabet (p. 181), as follow :
a = ABC e = 0BC
/? - AB<: = aEc
y = AC rj = abC
8 = Kbc 9=abc
282 THE LOGICAL INDEX. [CHAP.
It is obvious that each Greek letter appearing in the middle
column of the Index represents the presence of the corre
sponding combination, or rather its non-exclusion. Absence
of the Greek letter represents exclusion. Looking, for
instance, to No. 31, we learn that a = bc, an assertion of
the Vlth type, excludes all combinations except a, ft y,
specified at the top of the table, and 0, specified in the
centre column ; that is to say, the combinations consistent
with a = bc are ABC, ABr, AC, and abc. The principal
use of the Index, however, will be in the inverse direction,
to find the law corresponding to certain unexcluded com
binations. Taking, for instance, the combinations A&r, #BC,
a&C, their Greek signs are 5, e, r; ; to find their law, then,
we must look in the last table in the column headed (not-a),
(not-/?), (not-y), and in the line showing 8, c, 17, in the
middle column. We there find the two assertions A = c
- Kb of the IVth type (No. 230), as those corresponding
to the combinations in question.
3. With the aid of this Index we can infallibly and
rapidly solve all possible problems relating to three terms.
What assertion, for example, can we make which shall not
be contradictory to, and yet shall not be inferrible from, the
premise a = BC -\~abct Working out the combinations
unexcluded by this premise, we find them to be ABr, AC,
A&r, # BC, and abc, or /3, 7, S, e, 0. Of these, /? and 8 may be
removed simultaneously without wholly removing any letter,
that is to say without contradiction. Looking in the second
table of the Logical Index at No. 8 1, we find the proposition
A = AC (of type II.) as one which removes /3 and 8.
This is the required proposition which is, as it were, quite
neutral to the one assumed. In the same way we might
remove 7 and without contradiction, so that No. 34, or
Ab = be, of type XII., is another neutral proposition. It
xxviii.] THE LOGICAL INDEX. 283
may, I believe, be safely inferred that every proposition of
type XIII., that of the premise in question, will have at
least two propositions neutral to it, of types II. and XII.
respectively.
Suppose it be required, as a second instance, to define the
precise points of agreement and difference of two disput
ants, one of whom asserts that ( i ) Space = three-way
spread with points as elements (Henrici) ; while his oppo
nent holds that (2) Space = three-way spread, and at
the same time (3) Space has points as elements, but is not
known to be the only thing that has. The three assertions
are symbolised as below, the combinations excluded being
indicated by their Greek signs :
f/3
(,) A-Bdjf < 2 t = Ar."
(3) A = AC V
u
We see that the second disputant s assertions have a
logical force superior to that of the first by J, namely ,
which corresponds to assertion 5, or a~B = a~BC. In addition,
then, to all that the first asserts, he affirms that a three-way
spread which is not space has points as elements.
As a third instance of the power and flexibility of this
combinational logic, suppose it to be required to make an
exhaustive statement of all the inferences which can be
drawn from the theorem that Similiar figures (A) consist
of all whose corresponding angles are equal (B), and whose
corresponding sides are proportional (C). We proceed in
this way. The proposition is of the form A = BC, of the
Vlth type, and negatives ft y, 8, e. Any proposition, then,
which negatives one, two, or three of these combinations
284 THE LOGICAL INDEX. [CHAP.
will be inferrible from the theorem, but not equivalent to it.
All the possible inferences, therefore, are indicated in the
following table of Index Numbers, which, taken in con
nection with the Logical Index, sufficiently explains itself:
ft 65 /3 y 97 Py8 113
7 33 S 81 /3ye 105
8 17 ft 73 ft 8 e 89
e 9 y 8 49 y 8 6 57
7 e 4i
These fourteen assertions, which are all the possible
non-equivalent inferences, or the equivalents of these, were
detected by the Logical Index in a few minutes ; it would
be doubtfully possible, and in any case a most laborious
problem, to obtain an exhaustive statement of inferences
by any other method, if indeed any other method exists.
The want of space alone prevents my giving more abun
dant illustrations of the multitudes of logical problems
which may be solved infallibly and speedily by the use of
the Logical Index. It may be safely said that in four pages
of tables it gives the key to all possible logical questions,
relations or problems involving three distinct logical terms.
There is some possibility that the corresponding index
for the relations of four terms may some day be worked out,
as, when exhibited in like manner, it will occupy only one
volume of 1024 pages of a rather larger size than those of
this volume. There is no prospect whatever that the corre
sponding index for five terms will ever be exhaustively
published, since it would fill a library of 65,536 volumes,
each containing 1024 large pages. This fact will give some
faint idea of the possible number and complexity of logical
relations involving only a very moderate number of terms.
THE LOGICAL STAMP. 285
The Logical Stamp.
In my previous logical books x I described a Logical
Slate with five series of the combinations of the Logical
Alphabet engraved upon it. I first made such a slate in
May 1863, an d I have since frequently used it with much
saving of labour. The recent extensive introduction of
india-rubber printing stamps lately suggested to me the
idea that the most convenient method of obtaining the
logical combinations would be to stamp them on paper.
Two stamps producing the combinations of three and of
four terms as shown in columns IV. and V. of the Logical
Alphabet (p. 181), were made for me at a cost of about
eleven shillings.
They have been very successful, and leave nothing to -be
desired as regards the private study of logical problems.
One great advantage of the stamps over the slate is evident,
namely, that the work being done on paper can be preserved
for reference without copying.
The ABCD stamp can readily be utilised for problems of
five, six, or more terms. For six terms, for instance, it is
requisite to make four impressions and distinguish them by
writing EF, E/j eF, ef, above the respective impressions.
India-rubber stamps of any design can now be easily
ordered at all of the principal stationers.
1 Pure Logic, 1864, p. 68 ; Substitution of Similars, 1869, p. 54 ;
Elementary Lessons in Logic, 1870, p. 199 ; Principles of Science, 1874,
Vol. I., p. 1 10 ; New Editions, p. 96.
286
THE LOGICAL INDEX.
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XXVIII.]
THE LOGICAL INDEX.
287
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288
THE LOGICAL INDEX.
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XXVIII.]
THE LOGICAL INDEX.
289
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CHAPTER XXIX.
MISCELLANEOUS QUESTIONS AND PROBLEMS.
IT seems convenient to bring these Studies in Deductive
Logic to a close by adding a certain number of mixed
Questions and Problems, which may refer to any part of
logical doctrine. In some cases these questions pass the
bounds of formal and deductive logic. It is left to the
student to determine what part, if any, of the preceding
pages will assist him. To certain questions, however, are
appended references to other works where the proper
assistance will be found.
1. What may we expect to happen, in a logical point of
view, when an irresistible force meets with an infinite
resistance ?
2. If it is said to be false that what has the properties
A and B has those also of C and D, and vice versa, how
would you interpret this statement as affecting the possible
relations of A, B, C, and D ?
3. In how many ways may we in a purely logical point of
view contradict the assertion of Hobbes that Irresistible
might in the state of nature is right ? (See p. 182,
Question 7.) Specify the ways.
4. Analyse the logical import of the following passage
from the Wealth of Nations, book i. chapter viii. :
CHAP, xxix.] MISCELLANEOUS QUESTIONS. 291
It is not because one man keeps a coach, while his
neighbour walks afoot, that the one is rich, and the other
poor ; but because the one is rich, he keeps a coach, and
because the other is poor, he walks afoot.
5. In Harriet Martineau s Autobiography (vol. i. p. 355)
we are told that a certain lady, after receiving from Charles
Babbage a long explanation of his celebrated calculating
machine, terminated the conversation with the following
question : < Now, Mr. Babbage, there is only one thing more
that I want to know. If you put the question in wrong,
will the answer come out right?
If you think this question absurd, give distinct and
detailed reasons for thinking so, and reconcile them with
the fact that false premises may give a true conclusion.
6. Explain and illustrate the Aristotelian saying: Ex
veris fieri non potest ut falsum condudatur ; ex falsis contra
verum; and show some of its applications in the investiga
tion of nature. r R 1
7. A certain argument having been shown to involve
paralogism, inquire into the conditions under which this
failure does or does not tend to establish the contradictory
conclusion.
8. In a certain borough, on one occasion, the Liberal
party objected to 3624 voters, and the Conservative party
to 553 1 voters, the whole constituency being 10,000.
What is the least number of voters which can have been
objected to on both sides ? What is the greatest number ?
What is the most probable number, supposing the objections
to be made quite at haphazard ?
9. What is the logical, compared with the popular, in
terpretation of the injunction, This man is not on any
account to be ducked in the horse-pond. Explain the
difference.
292 MISCELLANEOUS QUESTIONS [CHAP.
10. What is the logical, compared with the popular, in
terpretation of the injunction, All persons are requested
not to discharge fireworks among the crowd around the
bonfire on the 5th November.
11. Because a horse is an animal, the head of a horse
is the head of an animal.
Examine the validity of this inference. Can you express
the reasoning syllogistically, or symbolically ? [E.]
12. Investigate the nature of the reasoning, good or bad,
involved in the four following examples :
(1) Elephants are stronger than horses; horses are stronger
than men; therefore elephants are stronger than
men. [E.]
(2) Alexander was the son of Philip ; therefore Philip
was the father of Alexander.
(3) As good kill a man as kill a good book ; for he that
kills a man does but kill a reasonable creature ;
but he that kills a good book kills Reason herself.
(4) Nay, look you, I know tis true ; for his father built a
chimney in my father s house, and the bricks are
alive at this day to testify it. [o.]
13. What methods underlie these inferences?
Because it froze last night, therefore the pools are
covered with ice.
During the retreat of the Ten Thousand a cutting north-
wind blew in the faces of the soldiers ; sacrifices were offered
to Boreas, and the severity of the wind immediately ceased,
which seemed a proof of the god s causation. [p.]
1 4. What method is employed in the following ?
* Brewster accidentally took an impression from a piece
of mother-of-pearl in a cement of resin and bees wax, and
finding the colours repeated upon the surface of the wax,
xxix.] . AND PROBLEMS. 293
proceeded to take other impressions in balsam, fusible
metal, lead, gum-arabic, etc., and always found the iri
descent colours the same. He thus proved that the chemical
nature is wholly a matter of indifference, and the form of
the surface is the condition of such colours.
15. What is the difference, logically, between the sen
tences : Leibnitz, a great philosopher, has said, etc. ; and
A great philosopher, Leibnitz, has said, etc. ? [c.]
1 6. What is successive induction, or induction by con
nection, as in the proof that ?z 2 = 1 + 3 + 54-
up to n of the odd numbers ? [H.]
(See Elementary Lessons in Logic, lesson xxvi. p. 220.)
1 7. Give an inductive proof that x n a n is divisible by
x a, when n is a whole number.
1 8. Can there be such a thing as a fallacy of simple in
spection, that is a fallacy which does not involve inference ?
(See Mill s System of Logic, book iv. chapter iii.)
1 9. Leibnitz says Knowledge is either obscure or clear.
The clear is again either confused or distinct : and the
distinct either adequate or inadequate; is further either
symbolical or intuitive ; and if it be at the same time both
adequate and intuitive, it is perfect.
Give an exhaustive classification of all possible kinds of
knowledge under the above conditions. (See Elementary
Lessons in Logic, lesson vii.)
20. Explain the logical meanings of the terms Genus,
Species, Difference, Property, and Accident, distinguishing the
meaning in extent and intent, and using for illustration the
varieties of things ABCD, ABG/, ABd), AB^, in which
A, B, C, D are terms denoting qualities, and c, d the
negatives of C, D.
21. Draw the inferences deducible from these data:
294 MISCELLANEOUS QUESTIONS [CHAP.
(1) A is the only antecedent always present when p is
present, and always absent when / is absent.
(2) A is an antecedent always present when p is present,
and always absent when p is absent.
(3) A is an antecedent frequently present when / is
present, and frequently absent when/ is absent. [p.]
22. Point out the exact nature of the relations between
the logical processes of Abstraction, Analysis, Synthesis,
and Generalisation.
23. What is the logical difference, if any, between nouns
Substantive and nouns Adjective ?
24. Is a Latin adjective used alone in the neuter an
adjective or a substantive ?
25. Is there self-contradiction in the assertion that know
ledge of what is outside my consciousness may be inside
my consciousness ?
26. Can absolute certainty be found in any conclusion (i)
inductively established, (2) deductively established? [E.]
27. Is there any distinction, and if so what, between a
general and an abstract notion, and is there a corresponding
difference between the names employed to express them ?
[E.]
28. It is a rule of syllogism that nothing can be inferred
from particular premises. How then can I infer from the
particular facts that some men have died, the universal
conclusion, All men die ? [E.]
29. Explain the limits of demonstrative science, and
examine the following statement : No matter of fact can
be matter of demonstration. [E.]
30. Distinguish Logical, Mathematical, and Physical
Quantity. [ K ]
31. Distinguish and exemplify Logical, Mathematical,
Metaphysical, Physical, and Moral Necessity. [E.]
xxix.] AND PROBLEMS. 295
32. When two phenomena are causally connected to
gether, can you always ascertain which is the cause and
which is the effect ? If so, how ? [L.]
33. Investigate how far or on what grounds our know
ledge of the following propositions approximates to
certainty :
* Nitric acid does not dissolve gold.
A distant fixed star is subject to gravity.
34. Consider from a logical point of view the assertion
that the increasing trade of Great Britain is caused by a
reform of the tariff. What kind of proof is applicable ?
35. A man having been shot through the heart imme
diately falls dead. Investigate the logical value of such a
fact as proving that all men shot through the heart will fall
dead. [L]
36. What do you understand by a working hypothesis ?
Under what conditions is it legitimate for an investigator to
employ hypothesis ? (Huxley. Mill s System of Logic, book
iii. chap. xiv. ; Principles of Science, chap, xxiii.) [L.]
37. State these arguments formally, and give their tech
nical designations :
(1) The thinking power does not belong to matter;
otherwise matter generally would exhibit it.
(2) Happiness is the reward of goodness ; and since all
do not desire a good life, all cannot obtain its
reward. [p.]
38. Why is it that with exactly the same amount of
evidence, both negative and positive, we did not reject the
assertion that there are black swans, while we should refuse
credence to any testimony which asserted that there were
men wearing their heads underneath their shoulders ? [P.]
296 MISCELLANEOUS QUESTIONS [CHAP.
39. What is the difference of meaning, if any, between
the propositions, This house was built by Jack, and
This is the house that Jack built ? (De Morgan, Third
Memoir on the Syllogism, loth page.)
40. Does the thesis that the ultimate premises in human
knowledge are the result of mental association affect the
nature and certainty of Logic, and if so, how ? [E.]
41. Define evidence. Distinguish intuitive, demonstra
tive, and probable evidence. [E.]
42. Explain : Certainty, therefore, has for its opposite,
uncertainty in one way impossibility in another. Uncer
tainty, in the language of logicians, is its contradictory
opposite impossibility, its contrary opposite. [P.]
43. Investigate the question whether the truth of a
statement is to be judged by the impression which it makes
upon those to whom it is addressed, by its literal corre
spondence with the belief of the person making it, or by any
other standard. [L.]
44. It has been pointed out by Ohm that reasoning to
the following effect occurs in some works on mathematics :
A magnitude required for the solution of a problem must
satisfy a particular equation, and as the magnitude x
satisfies this equation, it is therefore the magnitude required.
Examine the logical validity of this argument.
45. It is probable that Herodotus recorded only what he
heard concerning Ethiopia ; and it is not unlikely that most
that he heard was correct; so that we may accept his
account as true. Is this conclusion correct ?
46. There is a very strong probability that the eldest
child of a newly married couple will inherit the estate of
the husband. For, firstly, it is more probable than not that
there will be children of the marriage. Next, if a child is
born, it is more probable that it will be a son, for more boys
xxix.] AND PROBLEMS. 297
are born than girls. Thirdly, if a son be born, it will
probably survive its father. Examine this inference, [o.]
47. Consider the following argument : Many writings
that are not genuine were ascribed to Clemens Romanus ;
this Epistle was ascribed to him ; therefore this Epistle is
not genuine. [L.]
48. A student of geometry examines three isosceles
triangles and finds them agree in having equal angles at the
base; an excise officer examines three bottles of wine out
of a quantity imported and finds them agree in strength ;
a chemist analyses three specimens of a mineral and finds
them agree in composition : compare the inferences which
may be drawn in these cases.
49. What is the relation between classification and in
duction in general ? [L.]
50. When an experiment designed to produce a phe
nomenon fails to produce it, in how many ways may we
interpret or explain the meaning of the failure ? [L.]
51. In what ways may a physicist hope to explain away
an exceptional phenomenon ? [L.]
52. If we never find skins except as the teguments of
animals, we may safely conclude that animals cannot exist
without skins. If colour cannot exist by itself (airav yap
Xpw/xa ev o-GjpxTi), it follows that neither can anything that
is coloured exist without colour. So, if language without
thought is unreal, thought without language must also be so.
What do you think of this argument ? [o.]
53. If we are disposed to credit all that is told us, we
must believe in the existence not only of one, but of two or
three Napoleon Buonapartes ; if we admit nothing but what
is well authenticated, we shall be compelled to doubt the
existence of any. How, then, can we be called upon to
believe in the one Napoleon Buonaparte of history? [o.]
298 MISCELLANEOUS QUESTIONS [CHAP.
54. Brown asserts that all planets are spheroids; Jones
denies it ; Robinson asserts that Jones knows nothing about
the matter; Smith proves that in this case at least Robinson
is correct ; but Thomson refuses to accept the premises of
Smith s proof. What are the logical relations of the
parties ?
55. From the statement that blood-vessels are either
veins or arteries, does it follow logically that a blood-vessel,
if it be a vein, is not an artery ? Give your reasons.
5 6. It is asserted by some philosophers that all knowledge
is inductive in its origin, and it is generally allowed that
inductive inferences can be probable only ; if so, no know
ledge can be more than probably true. Can you, however,
adduce any instance of knowledge which is certainly true ?
In that case explain the difficulty which evidently arises.
57. Aut amat aut odit mulier ; nihil tertinm. If any
one takes upon himself simply to deny the truth of this
saying of Publius Syrus, in how many different ways may
the denial be interpreted ?
5 8. Explain the following apparent paradox : P thinks
of an object ; Q is absolutely ignorant of the size of that
object ; to him, therefore, the probability that the object is
greater than a cannon-ball is J. Again, being absolutely
ignorant about its size, he has no reason to believe it either
greater or less than a pea, the probability of either case
being -J. Hence to him it is infinitely improbable that
the object is intermediate in size between a pea and a
cannon-ball. [JOHN HOPKINSON, D.SC]
59. In defending a prisoner his counsel must either deny
that the deed committed is a crime, or he must deny that
the prisoner committed the deed ; therefore if the counsel
denies that the deed committed is a crime, he must admit
that the prisoner did commit the deed.
xxix.] AND PROBLEMS. 299
60. What do you understand by the logical proof of an
assertion ? Compare the logical meaning of the word proof
with any other meanings of the word known to you. [i.]
6 1. Can all kinds of propositions be exhibited in the
intensive as well as the extensive form ? Give reasons in
support of your answer ; and in the event of its being in
the negative, draw up a list distinguishing between those
kinds of propositions which can, and those which cannot,
be so exhibited. [L.]
62. Explain the meaning of the assertion that Induction
is the inverse process of Deduction.
63. Illustrate Mathematical Induction in its several kinds
or cases, and discuss its relation to induction in the physical
sciences.
64. What is the relation, if any, between the inductive
syllogism and the inductive methods employed in the
physical sciences ?
65. Estimate upon logical grounds the possibility of
establishing a school in which students should be rendered
capable of discovering the Laws of Nature. (Gore s Art of
Scientific Discovery^)
66. What precisely is meant by the Law of Continuity?
Point out the grounds and limits of its validity. (Life of
Sir W. Hamilton, p. 439 ; Principles of Science, chapter
xxvii.)
67. When the effects of three distinct causes are added
and mingled together, by what processes of experiment and
reasoning can we assign to each cause its separate effect ?
[a]
68. Under what circumstances are we to accept the
failure of an experiment or series of experiments as proving
the non-existence of the phenomenon intended to be
produced? (Principles of Science, chapter xix.) [L.]
300 MISCELLANEOUS QUESTIONS [CHAP.
69. Illustrate the scientific value of exceptional pheno
mena, and show in how many ways they may be disposed
of or reconciled with physical law. (Principles of Science,
chapter xxix.) [L.]
70. What is the difference between the causal and the
casual happening of events, if, as is generally allowed, not
even a dead leaf falls to the ground without sufficient causes
to determine the precise moment of its falling and the
precise spot upon which it will fall ?
71. Show by example that the logical copula does not
imply the notion of existence. [E.]
72. Investigate the question whether the functions of
affirmative and negative propositions in reasoning are
similar.
73. England is the richest country in the world, and has
a gold currency. Russia and India, in proportion to
population, are poor countries, and have little or no gold
currency. How far are such kinds of facts logically
sufficient to prove that a gold currency is the cause of a
nation s wealth? [i.]
74. If by two distinct methods of investigation you arrive
at the same conclusion, namely, that the currency of the
kingdom does not exceed one hundred millions sterling, but
it is afterwards discovered that one of the methods of
investigation involved fallacious reasoning, what would you
be inclined to infer about the other method of investigation?
M
75. A certain argument having been shown to involve
paralogism, inquire into the conditions under which this
failure does or does not tend to establish the contradictory
conclusion.
76. Investigate the logical, psychological, and moral
grounds of the saying, Qui s excuse, s accuse?
xxix.] AND PROBLEMS. 301
7 7. Taking the senses in which they most resemble one
another, distinguish between judgment, opinion, statement,
knowledge, fancy, conjecture, supposition, allegation. [E.]
78. Distinguish: truth, certainty, fact, opinion, proba
bility, evidence, conviction. [E.]
79. How far can the inconceivability of the opposite be
regarded as proof of the truth of any judgment ? [E.]
80. Right-angled and not-right-angled are contradictory
predicates; therefore, according to the law of Excluded
Middle, as the proposition All triangles are right-angled
is false, it must be true that all triangles are not right-
angled. But this also is false. Explain the above difficulty.
[E.]
8 1. Given that (i) whenever the statements a, b, x are
either all three true, or all three false, then the statement
c is false, and y is true, or else c is true, and y is false ;
(2) that whenever d, e, y are either all three true or all three
false, then the statement a is false, and x is true, or a is true,
and x is false. When can we infer from these premises that
either x or y is true ?
[Hugh MacColl, B.A., in Educational Times, question
6206. A solution was given by C. J. Monro, M.A.,
in the same paper for March 1880. The question
seems to mean What other conditions with those
given determine that either x or y is true ?]
82. De Morgan says (Fourth Memoir on the Syllogism,
p. 5) of the Laws of Thought : Every transgression of
these laws is an invalid inference ; every valid inference is
not a transgression of these laws. But I cannot admit that
everything which is not a transgression of these laws is a
valid inference. Investigate the logical relations between
these three assertions.
302 MISCELLANEOUS QUESTIONS [CHAP.
83. To what type of assertion do the premises of Darapti
belong ?
84. Give the converse, inverse, contrapositive, obverse,
and reciprocal propositions of the following :
(1) All parallelograms have their opposite angles equal.
(2) If P is greater than Q, then R will be greater
than S.
(3) Two triangles are congruent if the three sides of the
one are respectively equal to the three sides of the
other.
85. Why have some mathematicians been accustomed to
say that it is necessary to prove the converse of a mathematical
proposition ?
86. Where exactly lies the error of the Irishman, who
being charged with theft on the evidence of three witnesses
who had seen him stealing the article in question, proposed
to bring in his defence thirty witnesses who had not seen
him stealing it ?
87. Epimenides says that every statement of a Cretan is
a lie ; but Epimenides is a Cretan ; therefore what he says
is a lie ; therefore every statement of a Cretan is not a lie.
[E.]
88. If in saying that few strikes are beneficial, I feel
sure that the statement will be misinterpreted by those to
whom it is addressed, and that the statement no strikes
are beneficial, although not in my opinion literally true,
will more exactly convey to the hearers minds the im
pression which I believe to be true, ought I, having regard
to the moral obligation of speaking the truth, to use the
latter assertion or the former ?
89. I will go on/ said King James, I have been only
too indulgent. Indulgence ruined my father.
xxix.] AND PROBLEMS. 303
Express clearly the process of reasoning involved in this
utterance. Is it Induction ? or what ? [M.]
90. What is the relation, if any, between the inductive
syllogism and the inductive methods employed in the
physical sciences ?
91. Can the proposition, All A is all BJ be regarded as
representing a single act of thought ? (See Mind, vol. i.
P- 216.) [ K ]
92. Are the premises of Darapti given only in a
numerical form sufficient to prove the conclusion ?
93. Does it follow that, because some poetry is not in
verse, there must be some verse which is not poetry ? [H.]
94. Take the proposition, All sciences are useful, and
determine precisely what it affirms, what it denies, and what
it leaves doubtful, concerning the relations of the terms
science and useful thing.
95. Ascertain precisely how many distinct assertions
there are in the description of the conduct of the great
scholastic logician, John of Salisbury, after Thomas a
Becket had been murdered by his side : Tacitus sed mcerens,
continue se subduxit.
96. Can you represent equationally the contradiction
between Some Xs are not some Fs and There is one X
only and that is the only F ?
9 7. Which of the types of assertion involving three terms
are complete, in the sense of admitting no additional assertion
involving the same three terms without self-contradiction ?
98. If all things are either X or F, and all things are
either F or Z, what inference can you draw ?
99. Do the thirty-six moods of Hamilton s Syllogism with
quantified predicates (see table in Elementary Lessons,
p. 1 88 ; Thomson s Laws of Thought, section 103) comprise
all the possible weakened moods ?
304 MISCELLANEOUS QUESTIONS. [CHAP. xxix.
100. Is the student of logic, generally speaking, prepared
rapidly to analyse the two following propositions, and to say
whether or no they must be identical, if the identity of
synonyms be granted ?
(1) The suspicion of a nation is easily excited, as well
against its more civilised as against its more war
like neighbours, and such suspicion is with
difficulty removed.
(2) When we see a nation either backward to suspect
its neighbour, or apt to be satisfied by expla
nations, we may rely upon it that the neighbour is
neither the more civilised nor the more warlike of
the two.
[DE MORGAN, Third Memoir ; 1858, p. 181.]
1 01. Is the following proposition a definition or not ? Is
it on the matter or the form of the proposition that you
found your answer?
LOGICA EST ARS ARTIUM ET SCIENTIA SCIENTIARUM.
THE END.
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